card(Y \ Z) = card(E). ) . III. Denote by U the class of subsets of Y defined by. U = {Z|(Y â Ω. â²â². & Z is class of all β-powerful subsets of the set Y }. Assume that ...
ON SOME QUASIINVARIANT MEASURES AND DYNAMICAL SYSTEMS IN INFINITE DIMENSIONAL TOPOLOGICAL VECTOR SPACES
by
GOGI PANTSULAIA
THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHYSICAL AND MATHEMATICAL SCIENCES IN THE TBILISI STATE UNIVERSITY
c ⃝GOGI PANTSULAIA 2003 GEORGIAN TECHNICAL UNIVERSITY
DECEMBER 2003
CONTENT 0. Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1. Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2. Dynamical Systems and their Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3. Gaussian Measures in Infinite-Dimensional Topological Vector Spaces . . 51 4. Construction of Probability Borel Product-Measures in the Topological Vector Space RI by the Methods of the Haar Measure Theory . . . . . . . . . . . . 59 5. Existence of Invariant and Quasiinvariant Radon Measures in the Topological Vector Space RI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6. Invariant Borel Measures in the Topological Vector Space RN . . . . . . . . . . 85 7. Invariant Borel Measures in the Nonseparable Banach Space ℓ∞ . . . . . . . . 98 8. Vector Fields of velocities in the Infinite-Dimensional Topological Vector Space RN , which preserve the Measure µ . . . . . . . . . . . . . . . . . . . . . . . . . 107 9. Invariant Borel Probability Measures on the Unit Sphere in an Infinite-Dimensional Separable Complex-Valued Hilbert Space . . . . . 120 10. Invariant Borel Measures in Infinite-Dimensional Separable ComplexValued Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 11. Property of Essential Uniqueness for measures . . . . . . . . . . . . . . . . . . . . . . . 131 12. The structure of Elementary σ-Finite Invariant Measures . . . . . . . . . . . . 138 13. Duality of Measure and Category in the Infinite-Dimensional Separable Hilbert Space ℓ2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 14. A Formula of Substituting Variables in the Integral for Invariant Measures in the space ℓ2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 15. The Zero-One Law for Invariant Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 16. Absolutely Negligible and Absolutely Nonmeasurable Sets in the Topological vector space RN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 17. Independent Families of Sets and Some of their Applications to Measure Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 18. Separated Families of Probability Measures . . . . . . . . . . . . . . . . . . . . . . . . . . 190 Appendix: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 §1. Proof of the Michelsky-Sverkovsky Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 200 §2. A Formula of Substituting Variables in the Lebesgue Multiple Integral and Liouville’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 §3. The General Marginal Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
0.Preface
This book deals with certain aspects of the general theory of systems. We are primarily interested in infinite-dimensional systems which naturally appear as models of various physical, economic and social processes and are important from the standpoint of applications. The main technique developed in this work can be characterized as follows: we extensively utilize the methods of invariant (respectively, quasiinvariant) measures in infinite-dimensional topological vector spaces. Let us briefly survey the contents of the book. It consists of eighteen sections and the Appendix. 1. Basic Concepts. Some fundamental notions and important facts from set theory, general topology and measure theory are considered here. 2. Dynamical Systems and their Properties. The definition of a dynamical system which is due to Birkhoff (see [95]) is given. Some examples of dynamical systems are discussed in infinite-dimensional topological vector spaces. The proof of the famous result due to Krylov and Bogolyubov is considered. This result turns out to be a particular case of the Markov-Kakutani theorem on the existence of fixed points. Several important theorems from the ergodic theory are also considered (for example, the Poincar´e recurrence theorem, the Birkhoff theorem and so on). Consideration is also given of problems closely connected with the problem of constructing ergodic systems in infinite-dimensional topological vector spaces. In particular, an ergodic system is constructed when the base space coincides with some infinite-dimensional Banach space. 3. Gaussian Measures in Infinite-Dimensional Topological Vector Spaces. The history of development of this area of mathematics is given and the stimulating role of the discovery of Brownian motion is shown. 4. Construction of Probability Borel Product-Measures in the Topological Vector Space RI by the Methods of Haar Measure Theory. One example of a quasiinvariant probability Borel measure in the topological vector space RI , due to A. B. Kharazishvili (see [70]), is presented and further investigations in this direction are carried out. The construction mentioned above is generalized in the case of an arbitrary family of probability Borel measures on R with positive continuous distribution densities. This method leads to the conclusion that the Borel product-measure 2
defined on RI is quasiinvariant with respect to the everywhere dense vector subspace R(I) of RI . The Borel product-measure pointed out above can be considered as a specific realization (in a certain sense) of the Haar measure. We emphasize that the development of the product-measure theory in RI is based on some important methods in the Haar measure theory. Using the Kakutani theorem, the class of all translations is described, under which given Borel probability product-measures are quasiinvariant. It is proved that the vector space of all admissible translations of the canonical Gaussian measure coincides with the space ℓ2 (I), where ∑ ℓ2 (I) = {(xi )i∈I : (xi )i∈I ∈ RI & x2i < ∞}. i∈I
5. Existence of Invariant and Quasiinvariant Radon Measures in the Topological Vector Space RI . An example of a such nontrivial σ-finite Radon measure in R[0;1] is constructed which is invariant under everywhere dense in R[0;1] vector subspace of all real-valued polynomials defined in [0; 1] (see Theorem 2). Using the method of separating families of real functions for a parametric set I with card(I) > c, where c denotes the cardinality continuum, we solve the problem of the existence of Radon quasiinvariant probability measures in the topological space RI (see Theorem 4). It is proved,that for card(I) > c, the R(I) -quasiinvariant Borel probability measures defined in RI do not have the property of inner regularity. These results were obtained in [102]. 6. Invariant Borel Measures in the Topological Vector Space RN . The method of constructing nontrivial invariant σ-finite Borel measures in the vector topological space RN (where N denotes the set of all natural numbers) is given by starting from Lemma 1 due to A. B. Kharazishvili (see [69]). This section also contains a full description of the algebraic structure of all admissible translations of invariant measures defined in RN . Theorem 1 was obtained in [77]. Theorem 2 gives a full characterization of all those subgroups of the group RN under which a nontrivial invariant σ-finite Borel measure does exist. Theorem 2 was obtained in [77]. Further, it is proved that a translate-invariant Borel measure always exists in an arbitrary infinite-dimensional separable Banach space.
3
7. Invariant Borel Measures in the Nonseparable Banach Space ℓ∞ . The method of constructing a translate-invariant Borel measure defined on the nonseparable Banach space ℓ∞ is considered and a partial solution of the problem posed by Rogers [117] is presented. It is proved that the measure ν constructed in [34] has not the property p2a∗ . 8. Vector Fields of velocities in the Infinite-Dimensional Topological Vector Space RN which Preserve the Measure µ. We consider the vector fields of velocities defined in the infinite-dimensional topological vector space RN by a system of first order differential equations and preserving the measure µ which was constructed in Section 6. An analogue of Liouville’s theorem is discussed for the measure µ. 9. Invariant Borel Probability Measures on the Unit Sphere in an Infinite-Dimensional Separable Complex-Valued Hilbert Space. We discuss the method of constructing probability Borel measures defined on the surface of the unit sphere S 2 and invariant under a one-parameter group of unitary operators. The obtained results are applied to quantum mechanics. In particular, when the Hamiltonian of Schr˝odinger’s equation is a totally continuous Hermitian operator, a Borel probability measure is constructed on the unit sphere S ∗2 in the infinite-dimensional separable complex-valued Hilbert space L2 (R3 , C) such that the phase flow defined by the corresponding equation preserves this measure. Some results are obtained by using the Poincar´eCarath´eodory and Birkhoff-Chintchin theorems. 10. Invariant Borel Measures in Infinite-Dimensional Separable Complex-Valued Hilbert Space. For a concrete class of phase flows defined in infinite-dimensional separable complex-valued Hilbert spaces, nontrivial σ-finite Borel measures are constructed which are invariant under the action of the corresponding phase flows. 11. Property of Essential Uniqueness for measures. The definition of subsets with the property of essential uniqueness with respect to a class of some σ-finite measures is given. This class is described (in terms of Boolean algebras) by Theorem 1. The proof of the existence of a maximal measurable set (with respect to some class of measures) having the property of essential uniqueness is given by Theorem 2. The validity of Theorem 3 was first shown in [76]. In some concrete case, an example of a maximal measurable set with the property of essential uniqueness is constructed (see Theorem 4).
4
12. The Structure of Elementary σ-Finite Invariant Measures. The definition of an elementary part of an invariant measure is considered in [67]. The results of Lemma 1 and Lemma 2 are due to A. B. Kharazishvili. The notions of basis and dimension for elementary invariant measures in the case of an invariant measure space (E, G, S) are discussed. In particular, the dimensions defined there are calculated for some invariant measure spaces. 13. Duality of Measure and Category in the Infinite-Dimensional Separable Hilbert space ℓ2 . The duality between the measure λ∆ and the Baire category with respect to the sentence P0 (·) is studied in the infinite-dimensional separable Hilbert space ℓ2 . It is proved that an analogue of the Erd˝os-Sierpi´ nski duality principle is not valid for such objects. 14. A Formula for Substituting Variables in the Integral for Invariant Measures in the Separable Hilbert space ℓ2 . It is proved that the classical result of the substitution of a variable in the integral, being true for invariant measures in Euclidean spaces, is not valid in the Hilbert space ℓ2 . Necessary and sufficient conditions are found for groups of translations when this result holds true. 15. The Zero-One Law for Invariant Measures. The question of the existence of Borel subsets in the Hilbert space ℓ2 satisfying the zero-one law (under the pair (ν△ , ℓ2 )) is considered in this section with some related results. 16. Absolutely Negligible and Absolutely Nonmeasurable Sets in the Topological Vector Space RN . The following general question is posed: which properties of sets are preserved under translations in the space RN ? It is demonstrated that the Baire and Bernstein properties are preserved under translations in RN (see Theorem 1). Theorem 2 shows that the completion L of the σ-algebra of all Borel subsets of RN , with respect to an arbitrary nontrivial σ-finite Borel measure, is not RN -invariant. Methods of construction of absolutely negligible and absolutely nonmeasurable subsets in the space RN are also presented (see Theorems 4 and 7). 17. Independent Families of Sets and Some of their Applications to Measure Theory. In the case of an infinite base set E the question of the existence of a maximal (in the sense of cardinality) family of ℵ0 -independent subsets was considered by A. Tarski (see [66]). He proved that the maximal cardinality is equal to 2card(E) . This result found some interesting applications in different fields of mathematics. 5
For example, in general topology, the fact that in an arbitrary infinite base space card(E) E the cardinality of the class of all ultrafilters is equal to 22 was proved as a consequence of the above-mentioned result(see [67]). Using the method of independent families of subsets in the Euclidean space En , the maximal family (in the sense of cardinality) of pairwise orthogonal Dn -invariant extensions of the Lebesgue measure was first constructed in [67] . The notion of independence of a family of sets due to A. Tarski is generalized in this section. The maximal cardinality of such a family is determined by Theorem 1. An application of this method to the construction of a maximal family of orthogonal invariant extensions of the Haar measure defined on a locally compact topological group is given (see Theorem 2). The method of construction of nonelementary invariant extensions of the Haar measure is also given(Theorem 5). An analogous method yields a solution of the 9th Problem formulated in the monograph [67] (see Theorem 9). Most of the results presented in this section were obtained in [103], [106] and [107]. 18. Separated Families of Probability Measures. An interesting classification of separated families of probability measures is due to A.V. Skorokhod (see [55]). In connection with this classification, Theorem 1 is presented. The result 1) → 2) in Theorem 1 also belongs to Skorokhod. The converse result 2) → 1) is obtained in [104]. Lemma 2 contains a generalization of Ulam’s result and is due to A. B. Kharazishvili (see [68]). Theorem 2 contains a generalization of one result obtained by Z. S. Zerakidze in [134] (in this conection, see [104]). Appendix. The Appendix contains three subsections. The proof of the well-known Michelsky-Sverkovsky’s theorem with one of its applications are considered in Subsection I. The geometrical proof of the substitution of variables in the Lebesgue multiple integral essentially due to Ostrogradsky is given in Subsection II. Liouville’s theorem is also proved here. Subsection III deals with some typical questions connected with the General Marginal Problem. A method of construction of invariant measures in the infinite-dimensional vector space Rα is developed, by using the properties of some consistent systems of σ-finite measures.
6
1. Basic Concepts This work is concerned with some topics from the theory of invariant and quasiinvariant measures given in infinite-dimensional topological vector spaces and their applications to certain problems arising in various dynamical (in particular, ergodical) systems. Note that similar problems are treated in the following monographs and papers: [13], [14], [15], [22], [34], [50], [53]-[55], [61], [63], [66]-[75], [80], [81], [88] [94], [95], [97], [98],[114], [117], [118], [121], [122], [126], [128], [132], [133], [134]. In this preliminary section we introduce the notation and recall some elementary facts from set theory, general topology and measure theory. We will systematically use these facts in our further considerations. The symbol ZF denotes the so-called Zermelo-Fraenkel set theory which is one of the most important formal systems of axioms in the modern set theory (see [10]). The basic notions of the Zermelo-Fraenkel system are sets and the membership relation ∈ between them. The system ZF consists of several axioms which formalise various properties of sets in terms of the relation ∈. We are not going to point out here a precise list of these axioms and will actually work within the framework of the so-called ”naive set theory”. The symbol ZF C denotes the Zermelo-Fraenkel theory with the Axiom of Choice AC. In other words, ZF C is the theory ZF &AC, where AC stands for the Axiom of Choice. Presently, one is well aware of the fact that the ZF C theory forms the basis of all modern mathematics, i.e., almost all modern mathematical areas can be developed starting with the ZF C theory. The Axiom of Choice is a very powerful set-theoretical assertion which giving rise to many unusual and interesting consequences. Sometimes, in order to get some required result, we do not need the whole power of the Axiom of Choice. In such cases, it is sufficient to apply one of its weak forms. If x and X are any two sets, then the relation x ∈ Xmeans that x belongs to X. In this situation we also say that x is an element of X. The relation X ⊆ Y means that a set X is a subset of a set Y . The relation X ⊂ Y means that a set X is a proper subset of a set Y . If R(x) is a relation depending on an element x (or, in other words, R(x) is a property of an element x), then the symbol {x : R(x)} denotes the set (the family, the class) of all those elements x for which the relation R(x) holds. 7
The symbol ∅ denotes as usual the empty set, i.e., ∅ = {x : x ̸= x}. If X is any set, then the symbol P (X) denotes the family of all subsets of X, i.e., we have P (X) = {Y : Y ⊆ X}. The set P (X) is also called the power set of a given set X. Let X and Y be any two sets. Then as usual: X ∪ Y denotes the union of X and Y ; X ∩ Y denotes the intersection of X and Y ; X \ Y denotes the difference of X and Y ; X△Y denotes the symmetric difference of X and Y, i.e., X△Y = (X \ Y ) ∪ (Y \ X). Let X be an arbitrary nonempty set and ℜ be some class of its subsets. The class ℜ is called a ring of subsets of X if the following condition holds: (∀X1 )(∀X2 )(X1 ∈ ℜ & X2 ∈ ℜ → (X1 ∪X2 ∈ ℜ) & (X1 ∩X2 ∈ ℜ) &(X1 \X2 ∈ ℜ)). If the condition X ∈ ℜ holds too, then a ring ℜ is called an algebra of subsets of X. A ring (an algebra) ℜ is called a σ-ring ( σ-algebra) of subsets of X if (∀k)(∀Xk )(k ∈ N & Xk ∈ ℜ → ∪k∈N Xk ∈ ℜ). A measurable space is a pair (E, S), where E is a nonempty set and S is an σ-algebra of subsets of E. Each element X of S is called a measurable subset of E. We put X × Y = {(x, y) : x ∈ X, y ∈ Y }. The set X × Y is called the Cartesian product of given sets X and Y . In a similar way, applying recursion, one can define the Cartesian product X 1 × X2 × · · · × Xn of a finite family {X1 , X2 , · · · , Xn } of arbitrary sets. 8
If X is a set, then the symbol card(X) denotes the cardinality of X. Sometimes card(X) is also called the cardinal number of X. ω is the first infinite cardinal (ordinal) number. In fact, ω is the cardinality of the set N = {0, 1, 2, · · · , n, · · ·} of all natural numbers. Sometimes it is convenient to identify the sets ω and N . ω1 is the first uncountable cardinal (ordinal) number. Notice that, as usual, ω1 is identified with the set of all countable ordinal numbers (countable ordinals). Various ordinal numbers (ordinals) are denoted by α, β, γ, ξ, · · · . Let α be an ordinal number. We say that α is a limit ordinal if α = sup{β : β < ξ}. The cofinality of a limit ordinal α is the smallest ordinal ξ such that there exists a family {αι : ι < ξ} of ordinals satisfying the relation αι < α (ι < ξ), α = sup{αι : ι < ξ}. The cofinality of a limit ordinal α is denoted by the symbol cf (α). Clearly, we have the inequality cf (α) ≤ α for all limit ordinal numbers α. A limit ordinal number α is called a regular ordinal if cf (α) = α. A limit ordinal number α is called a singular ordinal if cf (α) ≤ α. For example, ω and ω1 are regular ordinals (cardinals) and ω ω is a singular ordinal (cardinal). 9
If k is an arbitrary infinite cardinal number, then the symbol k + denotes the smallest cardinal among all those cardinals which are strictly greater than k. For example, we have ω + = ω 1 , ω2 = (ω1 )+ , · · · . The symbol Q denotes the set of all rational numbers. The symbol R denotes the set of all real numbers. If the set R is equipped with the standard mathematical structures (order structure, algebraic structure, topological structure), then R is usually called the real line. Let n be a fixed natural number. The symbol Rn (respectively, S n ) denotes as usual the n-dimensional Euclidean space (respectively, the n-dimensional Euclidean unit sphere). Let X and Y be two sets. A binary relation between X and Y is an arbitrary subset G of the Cartesian product of X and Y , i.e., G ⊆ X × Y. In particular, if X = Y , then we say that G is a binary relation on the basic set X. For a binary relation G ⊆ X × Y , we put pr1 (G) = {x : (∃y)((x, y) ∈ G)}, pr2 (G) = {y : (∃x)((x, y) ∈ G)}. It is clear that G ⊆ pr1 (G) × pr2 (G). The Axiom of Dependent Choices is the following set-theoretical statement: If G is a binary relation on a nonempty set X and, for each element x of X, there exists an element y of X such that (x, y) ∈ G, then there exists a sequence (x0 , x1 , · · · , xn , · · ·) of elements of X such that (xn , xn+1 ) ∈ G for all n ∈ N. The Axiom of Dependent Choices is usually denoted by DC. Actually, the statement DC is a weak form of the Axiom of Choice which is completely sufficient for most areas of classical mathematics: geometry of a finite-dimensional Euclidean space, mathematical analysis of the real line, the Lebesgue measure theory and so on. The symbol c denotes the cardinality of the continuum, i.e., c = 2ω = card(R). The Continuum Hypothesis (shortly, CH) is the assertion that c = ω1 . K. G˝odel showed that the (ZF C) & (CH) theory is consistent since it is valid in the Constructible Universe (see [25]). On the other hand, P. Cohen showed that the (ZF C)&(¬CH) 10
theory is also consistent (see [13]). Hence the sentence CH is independent of the ZF C theory. A stronger form of the Continuum Hypothesis is the Generalized Continuum Hypothesis, denoted by GCH, which states that (∀k > ω)(2k = k + ). K. G¨odel also showed that the (ZF C) & (GCH) theory is consistent because it is valid in the Constructible Universe. From the moment it turned out that the Continuum Hypothesis was independent of ZF C, the issue of adding new axioms became really unbalanced. While the Continuum Hyphothesis is an extremely powerful assertion, perhaps even too strong, its negation is rather weak. Hence a natural necessity arose to find an appropriate axiom which even in the absence of the Continuum Hypothesis could give tools efficient enough for mathematical constructions. Martin’s axiom turned out to be a very good candidate to fill up this place. To formulate Martin’s Axiom, we need some notations and definitions. Let P be an arbitrary nonnempty set. A binary relation of G ⊆ P × P is called an equivalence relation on P if the following three conditions hold: 1) (p, p) ∈ G for all ellements p ∈ P ; 2) (p, q) ∈ G and (q, r) ∈ G imply (p, r) ∈ G; 3) (p, q) ∈ G implies (q, r) ∈ G. If G is an equivalence relation on P , then the pair (P, G) is called a set equipped with an equivalence relation. Obviously, if G is an equivalence relation on P , then we have a partition of P canonically associated with G. This partition consists of the sets G(p) (p ∈ P ), where G(p) denotes the section of G corresponding to an element p ∈ P ; in other words, G(p) = {q : (p, q) ∈ G}. Conversely, every partition of the set P canonically defines an equivalence relation on P . Let P be an arbitrary set and let G be a binary relation on P . We say that G is a partial order on P if the following three conditions hold: 1) (p, p) ∈ G for each element p of P ; 2) (p, q) ∈ G and (q, r) ∈ G imply (p, r) ∈ G; 3) (p, q) ∈ G and (q, r) ∈ G imply p = q. Suppose that G is a partial order on a set P . As usual, we write p ≼ q iff (p, q) ∈ G. The pair (P, ≼) is called a partially ordered set. 11
Let (P, ≼) be a partially ordered set. We say that a set D ⊆ P is dense in P if for each p ∈ P there is q ∈ D with p ≼ q. (This is actually the density in the topological sense for a suitable topology on P , viz., that having as a base all sets of the form {p : p ≽ q}, where q ∈ P ). A set D ⊆ P is called a subnet if for arbitrary two elements p1 ∈ P and p2 ∈ P there exists an element q ∈ D such that p1 ≼ q and p2 ≼ q. Two elements p and q are called compatible if there is an element r ∈ P such that p ≼ r and q ≼ r. Finally, we say that (P, ≼) satisfies the countable chain condition (or, simply, c.c.c.) if every uncountable subset of P contains at least two compatible elements. Martin’s Axiom usually denoted by M A is the following statement: If (P, ≼) is a partially ordered set satisfying c.c.c. and (Di )i∈I is a family of dense subsets in P with card(I) < 2ω , then there is a subnet D ⊆ P which intersects each Di . The Continuum Hypothesis readily implies Martin’s Axiom. Indeed, let us assume CH. Let (P, ≼) be any partially ordered set and let (Dn )n∈ω be any sequence of dense subsets of P . Then we can recursively construct an increasing sequence (pn )n∈ω of elements of P such that pn ∈ Dn for each n ∈ ω. Now, put D = {p ∈ P : (∃n ∈ ω)(pn ≼ p)}. Evidently, D is a subnet in P which intersects every Dn . Martin and Solovay proved that the statement (M A)&(¬CH) is consistent with ZF C (see, e.g. [59] or [122]). Another interesting set-theoretical assertion is the so-called Axiom of Determinacy (denoted by AD). This axiom can be formulated as a statement of the existence of a winning strategy for at least one of two players in a certain infinite game. (For more detailed information on the Axiom of Determinancy see the Appendix.) Let E be a basic set (in general, E is assumed to be infinite) and let T be a topological structure on E, i.e., a topology on E. As a rule, the pair (E, T ) is called a topological space. Let (E, T ) be an arbitrary topological space. We say that a set X ⊆ E is nowhere dense (in E) if int(cl(X)) = ∅. We say that a set Y ⊆ E is a first category subset of E if Y can be represented in the form Y = ∪n∈ω Yn , where all Yn (n ∈ ω) are nowhere dense subsets of E. We say that a set Z ⊆ E is a second category subset of E if Z is not a first category subset of E. Let us consider some facts from general topology which we will need to make use of in the sequel. It is understood that the reader knows some elementary facts and notions from this area of mathematics, for instance, such notions as a continuous mapping, separation axioms and their corresponding classes of topological spaces (Hausdorff spaces, regular spaces, completely regular spaces, 12
normal spaces and other spaces), quasi-compactness, metrizability of a topological space, completeness for metric spaces and so on (see, e.g., [25] or [64]). The notion of quasi-compactness is one of the most important ones among the topological notions listed above. Let us recall that a topological space E is quasi-compact if every open covering of E contains a finite subcovering of E. A space E is called compact if it is Hausdorff and quasi-compact at the same time. There are a lot of remarkable theorems connected with the notion of quasicompactness. The main one is, of course, the classical Tikhonov theorem. Theorem 1 (Tikhonov). The topological product of an arbitrary family of quasi-compact spaces is a quasi-compact space. Conversely, if the topological product of the family of nonempty spaces is quasi-compact, then each of these spaces is quasi-compact. The proof of this theorem can be found in [25] or [65]. Theorem 2 (Banach). Let E be any topological space and let (Vi )i∈I be any family of open first category subsets of E. Then the union V = ∪i∈I Vi is an open first category set, too. The proof of this theorem is given in [18]. From Theorem 2 we immediately obtain the following result. Theorem 3. Any topological space E can be represented as the union E = E1 ∪ E2 , where the set E1 is an open first category subset of E, the set E2 is a closed Baire subspace of E and E1 ∩ E2 = ∅. Similarly, any topological space E can be represented as the union E = E1∗ ∪ E2∗ , where the set E1∗ is a closed first category subspace of E, the set E2∗ is an open Baire subspace of E and E1∗ ∩ E2∗ = ∅. The following classical definition plays one of the main roles in this work. Let E be any topological space and let X be a subset of E. We say that the set X has the Baire property in E if X can be represented as X = (V ∪ Y ) \ Z, where V is an open subset of E, and Y and Z are first category subsets of E. It is easy to see that a set X ⊆ E has the Baire property if and only if there exist an open set V ⊆ E and a first category set P ⊆ E such that X = V △P. 13
The class of all sets which have the Baire property in E will be denoted by the symbol B(E) and the class of all first category subsets of E- by the symbol K(E). It is easy to prove the following theorem. Theorem 4. The class B(E) is a σ-algebra of subsets of E. This σ-algebra is generated by the union T (E) ∪ K(E), where T (E) is the topology of the space E. Let us recall that the Borel σ-algebra of a topological space E is the σalgebra generated by the topology T (E). The Borel σ-algebra of the topological space E is denoted by the symbol B(E). Every element X ∈ B(E) is called a Borel set in E. It is clear that the inclusion B(E) ⊆ B(E) always holds true. Let (E1 , S1 ) and (E2 , S2 ) be two measurable spaces and let f be a mapping from E1 into E2 . We say that f is a measurable mapping if (∀X)(X ∈ S2 → f −1 (X) ∈ S1 ). We say that a mapping f : E1 → E2 is a measurable isomorphism from E1 onto E2 if f is a measurable bijection and the inverse mapping f −1 is measurable, too. Let E1 and E2 be two topological spaces. We can consider two measurable spaces (E1 , B(E1 )) and (E2 , B(E2 )). Let f be a mapping from E1 into E2 . Let us recall that f is called a Borel mapping if f is a measurable mapping from E1 into E2 . We say that a mapping f : E1 → E2 has the Baire property if, for each open set V ⊆ E2 , the set f −1 (V ) has the Baire property in the space E1 . It is easy to show that the following statement is true. Theorem 5. If E1 and E2 are topological spaces and if f is a mapping from E1 into E2 , then the next three statements are equivalent: 1) the mapping f has the Baire property; 2) for any closed set Z ⊆ E2 the set f −1 (Z) has the Baire property in the space E1 ; 3) for any Borel set Z ⊆ E2 the set f −1 (Z) has the Baire property in the space E1 . 14
In particular, any Borel (hence, any continuous) mapping f : E1 → E2 has the Baire property. The following useful result is valid. Theorem 6. Let E1 and E2 be two topological spaces such that E2 satisfies the second countability axiom (i.e., there exists a countable base for T (E2 )). Let f be a mapping from E1 into E2 . Then the next two statements are equivalent: 1) the mapping f has the Baire property; 2) there exists a first category set Z ⊆ E1 such that the restriction of the mapping f to the set E1 \ Z is continuous. The proof of Theorem 6 can be found in [18]. Now we introduce some simple cardinal-valued functions describing various properties of topological spaces. We put ω(E) = inf{card(B) : B is a base of T (E)} + ω;
c(E) = sup{card(B) : B is a family of pairwise disjoint nonempty open sets in E}+ω;
d(E) = inf{card(X) : X is a dense subset of E} + ω;
πω(E) = inf{card(B) : B ⊆ T (E)\{∅} and B is coinitial in (T (E)\{∅}, ⊆)}+ω. These functions are usually referred to as: ω(E) is the weight of a space E; c(E) is the Suslin number of a space E; d(E) is the density of a space E; πω(E) is the π-weight of a spaceE. A topological space E is said to satisfy the Suslin condition if c(E) = ω. A family B ⊆ T (E) \ {∅} is said to be a π-base of the topological space E if B is a coinitial subset of (T (E) \ {∅}, ⊆) 15
The following inequalities are obvious: c(E) ≤ d(E) ≤ πω(E) ≤ ω(E). As is well known, for a metric space E all these inequalities become equalities (see, e.g., [18]). It is also easy to prove that the inequality card(E) ≤ 22
d(E)
holds for any Hausdorff topological space E. Example 1. Let α be any infinite cardinal. Let us consider the space Rα equipped with the product topology. It can be shown that this space satisfies the Suslin condition, i.e., c(Rα ) = ω. Moreover , if α = 2ω , then it can be shown that the space Rα is separable, i.e., d(Rα ) = ω. Indeed, in that case the space Rα can be identified with the space R[0,1] consisting of all real functions defined on [0, 1], equipped with pointwise convergence topology. Using, for instance, the classical Weierstrass theorem on approximation, we deduce that the countable set of all polynomials on [0, 1] with rational coefficients is dense in the space R[0,1] . If α > 2ω , then the topological space Rα is not separable but, as mentioned above, it always satisfies the Suslin condition. Next, we will introduce one notion which is important for our purposes and plays a remarkable role in the classical descriptive set theory. A topological space E is a Polish space if E is homeomorphic to a complete separable metric space. It is clear that any compact metric space is a Polish space. In particular, the Cantor discontinuum{0, 1}ω (where the set {0, 1} is equipped with the discrete topology) is a Polish space. The space ω ω = N ω , where N is equipped with the discrete topology, is another standard example of a Polish space. The space N ω is usually called the canonical Baire space. It is not difficult to prove that N ω is homeomorphic to the set of all irrational numbers in R. The following statement contains topological charactarisations of some important metric spaces. Theorem 7. The following three statements hold: 1) any nonempty Polish space is a continuous image of the space N ω ; 2) any nonempty compact metric space is a continuous image of the Cantor discontinuum {0, 1}ω ; 3) any separable metric space is topologically contained in the Hilbert cube [0, 1]ω . The proof of Theorem 7 can be found in [18]. 16
Let us consider some facts from measure theory. A measure of a set is a generalization of the notion of length of an interval, area of a plane figure, and volume of a three-dimensional body. The notion of a measurable set appeared in the real function theory during investigations and generalizations of the concept of an integral. A standard example of a measure is the Lebesgue measure on the real line R defined by Lebesgue in 1902. It extends the notion of interval length to a much wider class of subsets of R. This class of sets contains all Borel and all analytic subsets of the real line and many other subsets of R (see e.g.[89]). Let E be a nonempty basic set, let S be any algebra of subsets of E, and let µ be a function from S into the extended real line R = R ∪ {−∞, +∞} such that card(ran(µ) ∩ {−∞, +∞}) ≤ 1. We say that µ is finitely additive (or, simply, additive) if for every finite family {X1 , · · · , Xn } ⊆ S of pairwise disjoint sets, we have µ(∪ni=1 Xi ) =
n ∑
µ(Xi ).
i=1
Similarly, we say that µ is countably additive (or σ-additive) if for every countable family (Xi )i∈I ⊆ S of pairwise disjoint sets such that ∪i∈I Xi ∈ S we have µ(∪i∈I Xi ) =
∞ ∑
µ(Xi ).
i=1
Finally, we say that a function µ : S → R+ is a measure (defined on the algebra S) if the following conditions hold: a) µ(∅) = 0; b)(∀X ∈ S) (0 ≤ µ(X)); c) µ is σ-additive. A measure space is a triple (E, S, µ), where E is a nonempty basic set, S is an σ-algebra of subsets of E and µ is a measure on S. Let (E1 , S1 , µ1 ) and (E2 , S2 , µ2 ) be two measure spaces. The measures µ1 and µ2 are called isomorphic if there exists a measurable isomorphism from E1 onto E2 such that (∀X)(X ∈ S1 → µ1 (X) = µ2 (f (X))). A measure µ is called σ-finite if there exists a countable family (Xn )n∈N of subsets of S such that 17
∪n∈N Xn = E, (∀n ∈ N )(µ(Xn ) < +∞). A measure µ is called finite if (∀X ∈ S)(µ(X) < +∞). We say that a measure µ is a probability measure if µ(E) = 1. We say that a measure µ is complete if for every X ∈ S with µ(X) = 0 we have (∀Y )(Y ⊆ X → Y ∈ S). A measure µ is called nonzero (nontrivial) if µ(E) ̸= 0. A measure µ is called nonatomic if (∀X)(X ∈ S & µ(X) > 0 → (∃Y )(Y ⊂ X & Y ∈ S & 0 < µ(Y ) < µ(X)). A measure µ is called diffused (or continuous) if (∀x ∈ E)({x} ∈ S & µ({x}) = 0). For any nonzero measure µ we denote L(µ) = {X : (∃Y )(X ⊆ Y & Y ∈ S & µ(Y ) = 0)}. If S is a σ-algebra, then it is clear that the class L(µ) is a σ-ideal of subsets of E. Moreover, one can see that the measure µ is complete if and only if L(µ) ⊆ S. The members of the class L(µ) are called µ-measure zero sets or µ-negligible sets. For every measure ν, there exists the smallest (with respect to inclusion) complete measure µ extending ν. The measure µ is called the completion of the original measure µ. The following fact is fundamental for the whole measure theory. Theorem 8 (Carath´ eodory). Let µ be a measure on the algebra S of subsets of the basic set E. Then there exists a measure extending the original measure µ onto the σ-algebra generated by the algebra S. If the original measure µ is σ-finite, then this extension is unique. The proof of Theorem 8 is presented e.g. in [1], [3], [10], [26]. Suppose that µ is a measure on the algebra S of subsets of E. We define a real-valued function µ∗ on the class P (E) by the formula ∑ µ∗ = inf{ µ(Yn ) : (Yn )n∈N ⊆ S & X ⊆ ∪n∈N Yn }. n∈N
18
This function is called the outer measure associated with µ. We say that a subset Z of E is µ∗ -measurable if, for any set X ⊆ E, the following Carath´eodory condition holds: µ∗ (X ∩ Z) + µ∗ (X ∩ (E \ Z)) = µ∗ (X). It can be shown that the class of all µ∗ -measurable sets Z ⊆ E is a σ-algebra of subsets of E which contains the original algebra S. Moreover, the function µ∗ considered only on the class of all µ∗ -measurable sets is countably additive and therefore it is a measure. It also extends the original measure µ and is complete. These facts immediately imply Theorem 8. It is sometimes useful to consider, along with outer measures, their dual object, the so-called inner measure. Recall that if the original measure µ is defined on a σ-algebra, then a function µ∗ defined on the class P (E) by the formula µ∗ (X) = sup{µ(Y ) : Y ∈ S&Y ⊆ X} is called the inner measure associated with µ. A set X ⊆ E is called µ-massive (or a set with a full outer measure with respect to µ) if the equality µ∗ (E \ X) = 0 holds. It is easy to see that when we have a finite measure µ, a set X ⊆ E is µ-massive if and only if µ∗ (X) = µ(E). The complements of µ-massive subsets of E can be successfully applied to various problems connected with measure extensions. The following statement is valid. Theorem 9. Let (E, S, µ) be a measure space and let I be a σ-ideal of subsets of E such that (∀Y )(Y ∈ I → µ∗ (Y ) = 0). Then the formula ν(X△Y ) = µ(X) (X ∈ S, Y ∈ I) correctly defines a measure ν on the σ-algebra S(I) which extends the original measure µ, where S(I) denotes, as usual, the σ-algebra generated by the class S ∪ I. The proof of Theorem 9 is given in e.g. [18], [67].
19
Let E be again a basic set and let S be a σ-algebra of subsets of E. The pair (E, S) is usually called a measurable space . A real function f on E is called S-measurable if, for every Borel set X ⊆ R, the set f −1 (X) is in S. The simplest examples of S-measurable functions are characteristic functions IY defined by { 1, if x ∈ Y , IY (x) = , 0, if x ∈ / Y. where Y ∈ S. Any linear combination of several characteristic functions is called a step function. The following theorem shows that the class of all S-measurable functions is closed under all natural algebraic operations. Theorem 10. Let E be a basic set and let S be some σ-algebra of E. Suppose that Φ:R×R→R is a B(R × R)-measurable function, and suppose that f and g are two Smeasurable functions. Then the function h : E → R defined by the formula h(x) = Φ(f (x), g(x)) (x ∈ E) is S-measurable, too. The proof of Theorem 10 can be found e.g. in [18]. Let f : E → R be a function. We put f + (x) = max{f (x), 0} (x ∈ E), f − (x) = max{−f (x), 0} (x ∈ E). It is evident that 0 ≤ f +, 0 ≤ f −, f = f + − f −. From Theorem 10 it folllows at once that the function f is S-measurable if and only if both functions f + and f − are S-measurable. The class of S-measurable real functions is also closed under limit operations: if (fn )n∈N is any sequence of S-measurable functions and f = lim fn , g = inf fn , h = sup fn , n
n
20
n
then f , g and h are also S-measurable functions. Let (E, S, µ) be a measure space, let X be a µ-measurable subset of E, and let f : E → R be an S -measurable and nonnegative function. The µ-integral of the function f on the set X is defined by the formula ∫ X
∑ f dµ = sup{ inf(f |Xn )×µ(Xn ) : (Xn )n∈N ⊆ S is a partition of the set X}. n
∫ In many cases, the real number f dµ is also denoted by the symbol ∫ X f (x)dµ(x). X
Suppose now that f : E → R is any S-measurable function. Then we put ∫ ∫ ∫ + f dµ = f dµ − f − dµ X
X
X
if at least one of the integrals from the right-hand side of this equality is finite. If both integrals from the right-hand side of this equality are finite, then ∫ we say that the function f is integrable on the set X, and the real number f dµ is X
called the µ-integral of f on the set X. We say that the function f is µ-integrable if it is µ-integrable on the whole basic set E. The class of all µ-integrable functions on E is a Banach space with respect to the norm ∫ ||f || = (f + + f − )dµ. E
Of course, we identify here the functions which are equivalent with respect to the measure µ, i.e., we identify the functions which coincide almost everywhere (with respect to µ) on the basic set E. We expect of the reader to know some standard facts about integrable real functions such as the Lebesgue theorem on majorated convergence, the Fatou lemma, absolute continuity of integrals, etc. Let us take one more look at the notion of a measure space with a σ-finite measure. Suppose that (E, S, µ) is such a space and assume that µ(E) = ∞. Let (Xn )n∈N ⊆ S be a countable family of pairwise disjoint sets, such that ∪n∈N Xn = E and 0 < µ(Xn ) < +∞ for each n ∈ N . Let us consider the measure ν on the σ-algebra S defined by the formula 21
ν(X) =
∑ n∈N
1 µ(X ∩ Xn ) (X ∈ S). 2n+1 µ(Xn )
Observe that ν is a probability measure on S. If X is an arbitrary set from S, then ν(X) > 0 if and only if µ(X) > 0. In this case we say that the measures µ and ν are equivalent. Let I be a nonempty set of indices and suppose that ((Ei , Si , µi ))i∈I is a family of probability spaces. In the usual way, we define the product ∏ (E, S, µ) = (Ei , Si , µi ) i∈I
of this family∏of measure spaces. Let E = Ei be the Cartesian product of the family of basic sets (Ei )i∈I . i∈I
A subset X of E is called a rectangular set if it can be represented in the form ∏ X= Xi , i∈I
where Xi ∈ Si for every i ∈ I and the set {i ∈ I : Xi ̸= Ei } is finite. The family of all rectangular subsets X of E is denoted by the symbol P0 and the family of all finite unions of rectangular set - by the symbol P . Obviously, the family P is an algebra of subsets of E generated by P0 . We define a function µ : P0 −→ R by the formula
∏ ∏ µ( Xi ) = µi (Xi ). i∈I
i∈I
It is easy to verify that the function µ can be uniquely extended to a σadditive function on the algebra P . We denote this extension by the same symbol µ. Hence, by Theorem 8, the measure µ can be extended to the uniquely termined measure on the σ-algebra S = σ(P ). The latter σ-algebra is denoted ∏ ∏ by Si . The extended measure defined on the σ-algebra Si is denoted by i∈I ∏ i∈I µi and is called the product measure of the family of measures (µi )i∈I . The i∈I
product of the family ((Ei , Si , µi ))i∈I is the measure space ∏ ∏ ∏ ( Ei , Si , µi ). i∈I
i∈I
22
i∈I
Now, we recall another classical fact from measure theory, namely, the wellknown Fubini theorem, which reduces the integration of real functions on the product measure space to the integration on the factors. Theorem 11 (Fubini). Let (E1 , S1 , µ1 ) and (E2 , S2 , µ2 ) be two measure spaces with σ-finite measures and let (E, S, µ) = (E1 , S1 , µ1 ) × (E2 , S2 , µ2 ). Suppose that f : E −→ R is a µ-integrable function. Then: 1) for µ1 -almost every x ∈ E1 the function y −→ f (x, y) (y ∈ E2 ) is µ2 -integrable; 2) for µ2 -almost every y ∈ E2 the function x −→ f (x, y) (x ∈ E1 ) is µ1 -integrable; 3) the function
∫ x −→
f (x, y)dµ2 (y) E2
is µ1 -integrable and the function
∫
y −→
f (x, y)dµ1 (x) E1
is µ2 -integrable; 4)the equality ∫ ∫ ∫ ∫ ( f (x, y)dµ2 (y))dµ1 (x) = ( f (x, y)dµ1 (x))dµ2 (y) = E1 E2
E2 E1
∫
∫ f (x, y)d(µ1 (x) × µ2 (y))
= E1 ×E2
holds. Analogously, one can formulate and prove the Fubini theorem for the product of finitely many measure spaces with σ-finite measures. Let S be a given σ-algebra of subsets of a basic set E. A function ν : S −→ R 23
is called a signed measure on S if a) ν(∅) = 0; b) card(ran(ν) ∩ {−∞, +∞}) ≤ 1; c) ν is σ-additive. The next result actually reduces the notion of a signed measure to the usual notion of measure Theorem 12 (Hahn). Suppose that ν is a signed measure on the σ-algebra S of subsets of the basic set E. Then there exist two sets A ⊆ E and B ⊆ E, such that: 1) A ∩ B = ∅, A ∪ B = E; 2) A ∈ S, B ∈ S; 3) for every X ∈ S we have ν(A ∩ X) ≥ 0 and ν(B ∩ X) ≤ 0. The decomposition {A, B} of the basic set E, corresponding to the given signed measure ν, is called the Hahn decomposition of E with respect to ν. We define ν + (X) = ν(X ∩ A) (X ∈ S), ν − (X) = −ν(X ∩ B) (X ∈ S). It is obvious that ν + and ν − are ordinary measures on the σ-algebra S. Moreover, we have ν = ν+ − ν−. Hence any signed measure ν can be represented as a difference of two ordinary measures. This representation is called the Jordan decomposition of the signed measure ν. Note that the function |ν| defined by the formula |ν| = ν + + ν − is a measure on the σ-algebra S and is called the total variation of a given signed measure ν. We say that a signed measure ν is σ-finite if there exists a family (Xn )n∈N ⊆ S such that ∪n∈N Xn = E and |ν|(Xn ) < +∞ for every n ∈ N . Suppose now that (E, S, µ) is a measure space and ν is a signed measure on the same σ-algebra S. We say that ν is absolutery continuous with respect to µ if (∀X ∈ S)(µ(X) = 0 −→ ν(X) = 0).
24
The next result plays an important role in modern analysis and the probability theory. Theorem 13 (Radon-Nikodym). Suppose that (E, S, µ) is a measure space with a σ-finite measure and ν is a σ-finite signed measure on S, absolutery continuous with respect to µ. Then there exists a µ- measurable function f : E −→ R such that for every X ∈ S we have ∫ ν(X) = f dµ. X
Let E be an arbitrary topological space and let B(E) be the Borel σ-algebra of this space. We say that a measure µ is a Borel measure (on E) if the equality dom(µ) = B(E) holds. It is obvious that the specific properties of the original topological space E often imply the corresponding specific properties of the Borel measures on E. The following definition describes a very important class of Borel measures. Let E be an arbitrary Hausdorff topological space and let µ be a Borel measure on E. We say that the measure µ is a Radon measure if for each set X ∈ B(E) we have µ(X) = sup{µ(K) : K is compact in E & K ⊆ X}. We say that a Hausdorff topological space E is a Radon space if every σfinite Borel measure on E is a Radon measure. From the latter definition it immediately follows that any Borel subset of a Radon topological space is also a Radon space. The following proposition is essentially due to Ulam (for the proof see e.g.[6]). Theorem 14. Any Polish topological space E is a Radon space. A generalization of this result is considered in Section 19. Let (E, S, µ) be a measure space. A transformation g : E → E is called admissible (in the sense of invariance for measure µ )if (∀X)(X ∈ S → g(X) ∈ S & µ(g(X)) = µ(X)). A transformation h : E → E is called admissible (in the sense of quasiinvariance for measure µ ) if (∀X)(X ∈ S → h(X) ∈ S & (µ(X) = 0 ⇐⇒ µ(h(X)) = 0). 25
Note that an arbitrary admissible transformation in the sense of invariance is also an admissible transformation in the sense of quasiinvariance for an arbitrary nontrivial measure. In general, the converse proposition is not valid. Another important mathematical structure closely connected with measures is a group structure. Let E be a basic set and let G be some group of transformations of this set. Further, let D be some class of subsets of E. We say that the class D is G-invariant if (∀X ∈ D)(∀g ∈ G)(g(X) ∈ D). Let S be some σ-algebra of subsets of E and let µ be a measure defined on S. We say that the measure µ is G-quasiinvariant if: 1) S is a G-invariant class of subsets of E; 2) the class L(µ) of all µ-measure zero sets is also G-invariant. If instead of the condition 2) the stronger condition 3)(∀X ∈ S)(∀g ∈ G)(µ(X) = µ(g(X))) holds, then we say that the measure µ is G-invariant. We say that a G invariant measure µ has the property of metrical transitivity (or is metrically transitive) if, for an arbitrary set X ∈ dom(µ) with µ(X) > 0, there exists a countable family (gk )k∈N of transformations from G such that µ(E \ ∪k∈N gk (X)) = 0. A subset X ⊆ E is called an almost invariant set with respect to G-invariant (G-quasiinvariant) measure µ, if (∀g)(g ∈ G → µ(g(X)∆X) = 0). A structure (E, S, G, µ) is called an invariant measure space if the following conditions hold: 1) E is a nonempty set; 2) G is some group of transformations; 3) S is some G-invariant σ-algebra of subsets of E; 4) µ is a G-invariant measure defined on S. The following definition is important for the theory of invariant measures. Let (E, S1 , G, µ1 ) and (E, S2 , G, µ2 ) be two invariant measure spaces such that S1 ⊆ S2 . We say that µ2 is a G-invariant extension of µ1 if (∀X)(X ∈ S1 → µ2 (X) = µ2 (X)). 26
By using well-known Carath´eodory theorem, we can easily establish that a usual extension of an arbitrary invariant σ-finite measure, defined on some invariant σ-ring of subsets of the basic space, is an invariant measure, too. Below, we will consider some properties of products of invariant (quasiinvariant) measures. Let (Gi )i∈I be an arbitrary family of groups. Let ei be a neutral element of the ∑ group Gi (i ∈ I). The direct sum of the family of groups (Gi )i∈I is denoted by Gi and defined as i∈I
∑
Gi = {(gi )i∈I : (∀i)(i ∈ I → gi ∈ Gi & card({i : gi ̸= ei }) < ω}.
i∈I
By using the Fubini theorem, one can obtain the validity of the following assertion. Theorem 15. Let (Ei , Si , Gi , µi )i∈I be the family of invariant ∑ ∏ (quasiinvariGi µi is a ant) probability measure spaces. Then the product-measure i∈I
i∈I
invariant (quasiinvariant) probability measure. Observe that an analogue of Theorem 15 also holds for σ-finite invariant (quasiinvariant) measures when card(I) < ω. Let (G, .) be an arbitrary, locally compact topological group. For such a group, we have the well-known result, due to Haar, which states the existence (and, in a certain sense, the uniqueness) of a nonzero invariant Borel measure µ on G (see e.g. [38]). Speaking more exactly, the measure µ satisfies the following relations: 1) µ is a locally finite measure, i.e., for each point x ∈ G there exists an open neighbourhood V (x) of this point, such that µ(V (x)) < +∞; 2) µ is a Radon measure; 3) µ is invariant under all left translations of G, i.e., (∀X ∈ B(G))(∀g ∈ G)(µ(X) = µ(g · X)). The measure µ is called the (left) Haar measure on the group G. If the group G is σ-compact, then the Haar measure µ on G is a σ-finite measure. If the group G is compact, then the Haar measure µ is a finite measure, and we can obviously assume that, in this case, µ is a probability measure. 27
We remark here that the classical n-dimensional Lebesgue measure ln ( considered only on the Borel σ-algebra of the Euclidean space Rn ) is a very particular case of the Haar measure and coincides with the standard Borel measure on Rn .
28
2.Dynamical Systems and their Properties Mathematical modelling is a science whose main goal is to construct such mathematical models that can fully represent quantitative behaviour of an observed process. In a wide class of mathematical models we distinguish the so-called dynamical systems describing the behaviour of various physical, economic and social processes. Let us recall the classical notion of a dynamical system due to Birkhoff (see [95]). Let (X, ϱ) be a metric space, and let f (x, t)x∈X,t∈R be a family of transformations of X, i.e., the condition f (x, t) ∈ X holds for arbitrary t ∈ R and x∈X . We say that the family (f (x, t))x∈X,t∈R is a dynamical system if the family (f (x, t))x∈X,t∈R , considered as the mapping of two variables satisfies the following three conditions: 1) f (x, 0) = x for each element x ∈ X; 2) the mapping f is continuous with respect to the variables x and t; 3) if x ∈ X, t1 ∈ R and t2 ∈ R, then f (f (x, t1 ), t2 ) = f (x, t1 + t2 ). One can easily verify that the family (f (x, t))t∈R is a one-parameter group of transformations of (X, ϱ) for each element x ∈ X. The parameter t is understood as time. The transformation (f (x, t))t∈R : X → X is called the motion of a dynamical system. For concrete x ∈ X, the set {f (x, t) : t ∈ R} is called a trajectory of the corresponding motion. For concrete x ∈ X, the set {f (x, t) : T1 ≤ t2 ≤ T2 }, where −∞ ≤ T1 < T2 ≤ +∞, is called an arc of the trajectory denoted by f (x; T1 ; T2 ). If −∞ < T1 < T2 < +∞, then the arc is called finite. The positive number T2 − T1 is called the time length of the corresponding arc f (x; T1 ; T2 ). Now let us assume that an element x ∈ X is fixed. If the condition (∀t)(t ∈ R → f (x, t + τ ) = f (x, t)) 29
is fulfilled for some parameter τ ∈ R, then (f (x, t))t∈R is called periodical motion. The smallest positive number T for which the condition (∀t)(t ∈ R → f (x, t + T ) = f (x, t)) holds is called a period of motion (f (x, t))t∈R . Sometimes, we have situations where the condition (∀t)(t ∈ R → f (x, t) = x) holds for a concrete point x ∈ X. The point x, which corresponds to such a ”motion”, is called a rest point. Clearly, periodical motion that does not have the smallest positive period coincides with a rest point. Some motions (f (x, t))t∈R may also occur when (∀t1 )(∀t2 )(−∞ < t1 < t2 < ∞ → f (x, t1 ) ̸= f (x, t2 )). Therefore, in the case of a dynamical system, we can distinguish three topological types of trajectories : 1) a point; 2) a simple closed arc; 3) a continuous and one-to-one image of the open interval ]0; 1[. The corresponding types of motions are: 1) rest; 2) periodical motion; 3) nonperiodical motion. Remark 1. One can generalize the Birkhoff notion of a dynamical system for all topological spaces (X, τ ) in terms of the convergence generated by the topology τ . Example 1. The classical dynamical system theory considers motions defined by the system of ordinary differential equations dxi = Xi (x1 , x2 , · · · , xn ) (1 ≤ i ≤ n), dt where the right-hand sides are the continuous functions of x = (x1 , x2 , · · · , xn ) in some closed domain D of an n-dimensional Euclidean ”phase space.” A solution of the system is a set of functions of the form x1 = x1 (t), x2 = x2 (t), · · · , xn = xn (t) which reduces the system to the identity. A general solution of such a system contains n arbitrary constants. To specify a unique solution, we can write the initial conditions as (0)
(0)
x1 (0) = x1 , x2 (0) = x2 , · · · , xn (0) = x(0) n . Cauchy proved that the above system has a solution satisfying the initial conditions and this solution is unique if the right-hand sides of the system are 30
continuous and possess first order finite partial derivatives with respect to the (0) (0) variables x1 , x2 , · · · , xn for the values t = t0 , x1 = x1 , · · · , xn = xn . (0) (0) Let xi = fi (x1 , · · · , xn ; t) (i = 1, n ) denote a solution of the system. It can be considered as motion of the classical dynamical system. As is well known, an arbitrary solution can be extended, when t → ±∞, or it reaches, for the finite value t = T , the boundary of the domain D. An arbitrary solution (0) xi = fi (x1 , · · · , x(0) n ; t) (i = 1, n ) is a continuous function with respect to t and the coordinates of the initial point. Since the right-hand sides of this system do not depend on the parameter t, as the motion started at the point x at the instant t1 reaches the point x1 , the motion started at x1 at the instant t2 reaches the point x2 . Hence the motion reaches the point x2 at the instant t1 + t2 . Now it is not difficult to verify that the function f (x, t) defined by f (x, t) = (fi (x1 , x2 , · · · , xn ; t))1≤i≤n is a dynamical system in the sense of Birkhoff. Example 2. Let (X, ϱ) be an arbitrary metric space. Let us put (∀x)(∀t)(x ∈ X & t ∈ R → f (x, t) = x). Note that every motion of the given dynamical system is a rest point. Example 3. Let Rn be the n-dimensional Euclidean space. Put (∀x)(∀t)(x ∈ Rn \ {0} & t ∈ R → f (x, t) = et · x). It is obviously that (f (x, t))t∈R is a nonperiodical motion for all x ∈ Rn \{0}. Example 4. Let us denote by R the real axis. Put (∀x)(∀t)(x ∈ R & t ∈ R → f (x, t) = x + at), where a ∈ R \ {0}. It is easy to see that (f (x, t))x∈R,t∈R is an example of a dynamical system whose every motion is nonperiodical. Example 5. Now, let C be the two-dimensional space of complex numbers. Put (∀z)(∀t)(z ∈ C & t ∈ R → f (z, t) = z · eit ). Similar calculations show us that (f (x, t))x∈X,t∈R is a dynamical system whose every motion is periodical. Example 6. If (X, ||.||) is an arbitrary infinite-dimensional Banach space, then the system (f (x, t))x∈X,t∈R defined by (∀x)(∀t)(x ∈ X & t ∈ R → f (x, t) = x + a · t), 31
where a ∈ X and ||a|| = ̸ 0, is an example of a dynamical system whose every motion is nonperiodical. Let (f (x, t))x∈X,t∈R be some dynamical system defined in a metric space (X, ϱ). The Borel measure ν defined in X is called invariant under the group of all motions of the dynamical system (f (x, t)x∈X,t∈R if for an arbitrary νmeasurable set A ⊆ X the relation (∀t)(t ∈ R → ν(f (A, t)) = ν(A)) holds. N. Krylov and N. Bogolyubov constructed invariant probability Borel measures for dynamical systems defined in compact metrizable spaces (see [95]). Their construction is essentially based on some well-known facts of measure theory, which will be formulated below. Recall that the family (νn )n∈N of probability Borel measures defined in a compact metric space X is weakly convergent to the measure ν if the condition ∫ ∫ lim φ(x)dνn = φ(x)dν n→∞
X
X
holds for an arbitrary continuous function φ : X → R. It is well known that the space C(X) of all continuous functions defined on X and equipped with the metric ϱ, where (∀φ1 )(∀φ2 )(φ1 ∈ C(X) & φ2 ∈ C(X) → ϱ(φ1 , φ2 ) = max |φ1 (x) − φ2 (x)| ), x∈X
is a separable metric space. Note that a functional A : C(X) → R defined by ∫ (∀φ)(φ ∈ C(X) → A(φ) =
φ(x)dν(x)), X
where ν is some fixed probability Borel measure in (X, ϱ), is a normed linear positively definite functional. Riesz and Radon proved that the converse proposition is also true. In particular, the following corollary is valid. Theorem 1. Let X be a compact metric space. Let C(X) be the space of all continuous functions defined on X. Then for an arbitrary normed linear positively definite functional A : C(X) → R, there exists a probability Borel measure ν in X such that 32
∫ (∀φ)(φ ∈ C(X) → A(φ) =
φ(x)dν). X
The proof of Theorem 1 can be found in [6]. We say that a family (µn )n∈N of probability Borel measures is weakly compact if there exists a subsequence (µnk )k∈N weakly converging to some probability measure µ in X. The following proposition is due to Helly (see [6]). Theorem 2 . An arbitrary countable family of Borel probability measures defined in an arbitrary compact metric space is weakly compact. Below, we will consider the proof of the famous result due to N. Krylov and N. Bogolyubov. Theorem 3. Let (f (x, t))x∈X,t∈R be a dynamical system defined in a compact metric space X. Then there exists a probability Borel measure which is invariant under the group of all motions (f (x, t))t∈R (x ∈ X). Proof. Let m be an arbitrary probability Borel measure defined in X. For concrete τ ∈ R \ {0} and φ ∈ C(X) we set 1 Aτ (φ) = τ
∫τ
∫ dt
0
φ(f (x, t))dm(x). X
By Theorem 1, we easily obtain the existence of a probability Borel measure mτ such that 1 τ
∫τ
∫ dt
0
∫ φ(f (x, t))dm(x) =
X
φ(x)dmτ (x). X
Hence we conclude that the family of probability Borel measures (mτ )τ ∈R\{0} is weakly compact, and by Theorem 5 we can construct a sequence (τn )n∈N with lim τn = ∞ and find a probability measure µ such that the family (mτn )n∈N n→∞ of probability Borel measures is weakly convergent to the measure µ. Therefore, the condition 1 n→∞ τn
∫τn
lim
∫ dt
0
∫ φ(f (x, t))dm(x) = lim
n→∞
X
X
33
∫ φ(x)dmτn (x) =
φ(x)dµ(x). X
is fulfilled for an arbitrary continuous real function φ on X. We have to show that the measure µ is invariant under the group of all motions (f (x, t))t∈R (x ∈ X). Indeed, the measure µ is invariant under the group of all motions if and only if the equality ∫ ∫ φ(x)dµ(x) = φ(f (x, t0 ))dµ(x) X
X
holds for an arbitrary parameter t0 ∈ R and for an arbitrary continuous function φ ∈ C(X). The above relation remains true when φ = φG is the indicator-function of an arbitrary open set G ⊆ X. Hence, having the validity of the relation ∫ ∫ φG (x)dµ(x) = φG (f (x, t0 ))dµ(x) X
X
for an arbitrary open set G, we get ∫ ∫ φG (x)dµ(x) = µ(G) = φG (f (x, t0 ))dµ(x) = µ(f (G, −t0 )), X
X
i.e., µ(G) = µ(f (G, −t0 )). Note that this relation can be easily extended to the class of all measurable subsets of X. Conversely, the validity of the condition µ(G) = µ(f (G, −t0 )) for an arbitrary measurable subset G ∫⊆ X and for an arbitrary parameter t0 ∈ R ∫ implies the equality of the integrals φ(x)dµ(x) and φ(f (x, t0 ))dµ(x). This X
X
equality is equivalent to the condition 1 lim n→∞ τn
∫τn
∫ dt
0
1 φ(f (x, t))dm(x) = lim n→∞ τn
∫τn 0
X
∫ dt
φ(f (x, t0 + t))dm(x). X
By the Fubini theorem, the functions under the limit operation can be rewritten as follows: ∫ dm(x) X
1 τn
∫τn φ(f (x, t))dt 0
34
and ∫
1 dm(x) τn
∫τn φ(f (x, t0 + t))dt. 0
X
Obviously, we get the following estimation: |
1 τn
∫τn φ(f (x, t))dt −
1 τn
∫τn
0
≤
1 {| τn
φ(f (x, t + t0 ))dt| ≤ 0
∫t0
τ∫ n +t0
φ(f (x, t))dt| + |
φ(f (x, t)dt|} ≤
2|t0 |M , τn
τn
0
where M is the maximum of |φ| on X. Indeed, M < ∞ because φ is a continuous real function defined on the compact metric space X. Hence there exists a finite limit 1 lim n→∞ τn
∫τn
∫ dt
0
φ(f (x, t))dm(x). X
The latter relation shows that for a sufficiently large natural number n, the module of the difference given above becomes less than an arbitrary small positive number ϵ, which completes the proof of the theorem. In general cases, the behaviour of some dynamical systems can be characterised by the following important theorem. Theorem 4 (Markov-Kakutani). Let E be a Hausdorff topological vector space over R, let T be a nonempty compact convex set in E,let U be a set of linear transformations of the space E into itself such that the conditions (∀u)(∀v)(u ∈ U & v ∈ U → u ◦ v = v ◦ u), (∀u)(u ∈ U → u(T ) ⊆ T ), (∀u)(u ∈ U → the restriction of u to T is continuous) hold. Then there exists a point x0 ∈ T such that (∀u)(u ∈ U → u(x0 ) = x0 ).
35
The proof of Theorem 4 can be found e.g. in [67]. Note that the Krylov-Bogolyubov theorem is a direct consequence of the Markov-Kakutani theorem. Indeed, let us denote by C(X) a vector space of all continuous functions f :X→R equipped with the usual norm ||f || = max |f (x)|. x∈X
Obviously, C(X) is a vector space over the field R of all real numbers. On the other hand, C(X) is a Banach space under the norm defined above. Taking into account the definition f ≤ f ′ − (∀x)(x ∈ X → f (x) ≤ f ′ (x)), we conclude that the space C(X) is an ordered Banach space. Let E be the space of all linear continuous functionals defined in C(X). Note that E is a Hausdorff locally convex topological vector space over the field R. Let us denote by T the subset of E consisting of linear functionals Φ : C(X) → R which satisfy the two conditions: a) (∀φ)(∀φ1 )(φ ∈ C(X) & φ1 ∈ C(X) & φ ≤ φ1 → Φ(φ) ≤ Φ(φ1 )), b) Φ(IX ) = 1, where IX is an indicator-function of the set X. It is easy to see that T is a convex closed subset in E. Let us show that T is a compact subset in E. If we set φ+ (x) = max(φ(x), 0) & x∈X
−
φ (x) = min(φ(x), 0) x∈X
for all functions φ ∈ C(X), then |Φ(φ)| = |Φ(φ+ ) + Φ(φ− )| ≤ Φ(φ+ ) + Φ(−φ− ) ≤ 2||φ||. This means that Φ∈
∏
[−2||φ||, 2||φ||]φ ,
φ∈C(X)
i.e., T ⊆
∏
[−2||φ||, 2||φ||]φ .
φ∈C(X)
36
By the Tikhonov theorem, the set ∏ [−2||φ||, 2||φ||]φ φ∈C(X)
is compact in RC(X) . Hence, taking into account the pointwise of convergence in E and in RC(X) , we conclude that T is compact in E. Now let us verify that the set T is not an empty one. Indeed, if x0 is an arbitrary element of X, then for all φ ∈ C(X) we can put Φ0 (φ) = Φ(φ(x0 )). It is clear that Φ0 is a linear positively definite continuous functional on C(X) and Φ0 (IX ) = 1. Since Φ0 ∈ T, we conclude that T is not empty. Let us define φt : X → R by the formula φt (z) = φ(f (z, t)) (z ∈ X), where t ∈ R, z ∈ X and φ ∈ C(X). Clearly, the relation (∀t)(t ∈ R → φt ∈ C(X)) is valid. Consider a functional Φt : C(X) → C(X) (t ∈ R) defined by (∀t)(t ∈ R & φ ∈ C(X) → Φt (φ) = Φ(φt )). Define also the mapping Ut : E → E for an arbitrary parameter t ∈ R by the formula (∀Φ)(Φ ∈ E → Ut (Φ) = Φt ). Now it is easy to verify that the following conditions are satisfied: a) (∀t)(t ∈ R → Ut is a continuous linear mapping of E into itself); b) (∀t)(t ∈ R → Ut (T ) ⊆ T ); c) (∀t1 )(∀t2 )(t1 ∈ R & t2 ∈ R → Ut1 ◦ Ut2 = Ut2 ◦ Ut1 ); 37
d) The family (Ut )t∈R of mappings is a group with respect to the usual composition operation, being isomorphic to the one-parameter group (f (z, t))t∈R (z ∈ E) . Accordingly, we can imply the Markov-Kakutani theorem for T and (Ut )t∈R . Let Φ be an element in T such that (∀t)(t ∈ R → Ut (Φ) = Φ ). The latter condition shows that (∀t)(t ∈ R → Φt = Φ ), i.e., (∀t)(∀φ)(t ∈ R & φ ∈ C(X) → Φ(φt ) = Φ(φ)). Since Φ is a normed linear positively definite functional, the Riesz-Radon theorem implies that there exists a probability Borel measure µ such that ∫ (∀φ)(φ ∈ C(X) → Φ(φ) = φ(x)dµ). X
On the other hand, the equality Φ(gt ) = Φ(g) (t ∈ R) means that the condition ∫ ∫ (∀φ)(φ ∈ C(X) → φ(f (x, t)dµ(x) = φ(x)dµ(x) ). X
X
holds. Thus it remains to repeat the final part of the proof of Theorem 3. Remark 2. Note that for the dynamical systems constructed in Examples 1, 2, 4, 5, 6, we cannot imply the Krylov-Bogolyubov theorem (also the MarkovKakutani theorem), but there exist non-trivial σ-finite Borel measures which are invariant under the group of all motions of the corresponding dynamical systems. It is clear that such a measure does not exist for the dynamical system constructed in Example 3. On the other hand, we can indicate a probability Borel measure (for example, a canonical Gaussian probability measure in R2 ) which satisfies the condition (∀A)(∀t)(A ∈ B(X) & t ∈ R → (µ(f (A, t)) = 0 ↔ µ(A) = 0)). A measure satisfying the above condition is called quasiinvariant under the group of all motions generated by the dynamical system f (x, t)x∈X,t∈R .
38
Now, we are going to discuss the following important notion. The dynamical system (f (x, t))x∈X,t∈R with an invariant (or quasiinvariant) measure µ defined on some invariant (under the group of motions) σ-algebra of subsets of X, is called an ergodic system. Remark 3. Various equivalent definitions of the ergodicity of dynamical systems are presently available (for example, see [35], [36], [95], [98]). By using similar methods of the theory of invariant and quasiinvariant measures, we can obtain the following classical results due to Poincar´e and Carath´eodory. Theorem 5 (Recurrence of a Subset). Let µ be a probability measure defined in a metric compact space X and being invariant under the group of all motions of the dynamical system (f (x, t))x∈X,t∈R . If A ∈ dom(µ) and µ(A) > 0, then there exists an instant t such that 1) |t| > 0 . 2) µ(A ∩ f (A, t)) > 0. Proof. We set B = {x : x ∈ A & (∀t)(|t| > 0 → f (x, t) ∈ / A)}. One can easily verify that B ∈ B(X). Let us show that µ(B) = 0. We begin by noting that f (B, t1 ) ∩ f (B, t2 ) = ∅ for all distinct t1 , t2 ∈ R. Indeed, if we assume the contrary, then f (B, t1 ) ∩ f (B, t2 ) ̸= ∅ for some dinstinct nonzero t1 , t2 ∈ R. Let y ∈ f (B, t1 ) ∩ f (B, t2 ). Then y ∈ f (f (B, t1 ), −t1 )∩f (f (B, t2 ), −t1 ) = f (B, 0)∩f (B, t2 −t1 ) = B∩f (B, t2 −t1 ). But the last relation means that for y ∈ B there exists an element x ∈ B such that y = f (x, t2 − t1 ), i.e., f (y, t1 − t2 ) = f (x, 0) = x. Thus,we obtain a contradiction with the definition of the set B. 39
Now, if we consider an infinite sequence of dinstinct instants (tk )k∈N , then (f (B, tk ))k∈N will be an infinite family of disjoint images of the set B. If we assume that µ(B) > 0, then we obtain a contradiction to the finiteness of the measure µ. Indeed, on the one hand we have, µ(X) = 1 < +∞. On the other hand, µ(X) ≥ µ(∪i∈N f (B, tk )) =
∑
µ(f (B, tk )) = +∞.
k∈N
We obtain a contradiction, and the validity of the relation µ(B) = 0 is proved. Consequently, µ(X \ B) = 1, i.e., µ{x : x ∈ A & (∃t)(|t| > 0 → f (x, t) ∈ A)} = µ(X) = 1. The proof is completed. We say that a point x is stable in the sense of Poisson if for an arbitrary neighbourhood U of x and for arbitrary T > 0 there exist t1 > T and t2 < −T such that f (x, t1 ) ∈ U and f (x, t2 ) ∈ U , respectively. Theorem 6 (Recurrence of Points). Let X be a metric compact space, and let µ be a Borel probability measure being invariant under the group of all motions of the dynamical system (f (x, t))x∈X,t∈R . Then µ-almost every point of X is stable in the sense of Poisson; moreover, (∀T )(∀A)(T > 0 & A ∈ B(X) → µ{x : x ∈ A & (∀t)(|t| > T & f (x, t) ∈ / A)} = 0), where B(X) denotes the σ-algebra of all Borel subsets of X. Proof. Let T be an arbitrary positive number. Let U be an arbitrary open set in X. Note that U can be considered as a neighbourhood of its points. Define B = {x : x ∈ U & (∀t)(|t| > T → f (x, t) ∈ / U )}. First, let us show that (∀t1 )(∀t2 )(|t1 − t2 | > T → f (B, t1 ) ∩ f (B, t2 ) = ∅). Indeed, if we assume the contrary, then there exist t1 ∈ R, t2 ∈ R & t1 − t2 > T , such that f (B, t1 ) ∩ f (B, t2 ) ̸= ∅. Hence f (f (B, t1 ), −t1 ) ∩ f ((B, t2 ), −t1 ) ̸= ∅ → B ∩ f (B, t2 − t1 ) ̸= ∅. 40
If y ∈ B ∩ f (B, t2 − t1 ), then there exists x ∈ B such that y = f (x, t2 − t1 ). Consequently, x = f (y, t1 − t2 ) and we obtain a contradiction with the condition y ∈ B & (∀t)(|t| > T → f (x, t) ∈ / A), so that f (y, t1 − t2 ) ∈ U and |t1 − t2 | > T. Now, we are going to show that µ(B) = 0. Indeed, if we assume the contrary, then (f (B, 2kT ))k∈Z is an infinite countable disjoint family of images of the set B, and from the invariance of µ we obtain ∑ 1 = µ(X) ≥ µ(∪k∈Z f (B, 2kT )) = µ(f (B, 2kT )) = +∞. k∈Z
This is a contradiction, and Theorem 6 is proved. The following proposition is of some interest. Theorem 7. Let (X, ρ) be a metric space equipped with the discrete metric ρ defined by { 1, if x ̸= y, (x ∈ X, y ∈ X). ρ(x, y) = 0, if x = y, If the cardinality of X is finite, then an arbitrary dynamical system (f (x, t))x∈X,t∈Z is periodical. Proof. Obviously, X is a metric compact space. Let us consider a classical probability measure µ defined in X and being invariant under the group of all motions of (f (x, t))x∈X,t∈R . Let X = (xi )1≤i≤n . By Theorem 5, we obtain the existence of an instant Ti such that f (xi , Ti ) = xi (1 ≤ i ≤ n). Let us show that f (xi , kTi ) = xi for arbitrary k ∈ Z & k ≥ 0. Indeed, f (xi , kTi ) = f ((· · · f ((f (xi , Ti ), Ti ), · · ·), Ti ) = xi . Analogously, the above-mentioned result is valid when k ∈ Z & k < 0. Let T ∗ be the least common multiple of numbers (Ti )1≤i≤n . Clearly, (f (x, t))x∈X1 ,t∈Z is a periodical dynamical system with period T ∗ . We say that a dynamical system defined on finite X is homogeneous if points of an arbitrary trajectory have equal periods. Remark 4. Let (f (x, t)x∈X,t∈Z is a homogeneous dynamical system defined on the finite set X∑and let (Xk )1≤k≤q be the family of all trajectories. It is clear that the relation 1≤k≤q |Xk | = |X| = n holds. 41
On the one hand, we have l.c.m(|X1∗ |, · · · , |Xq∗ |) ≤
∏
|Xk |,
1≤k≤q
where l.c.m.(|X1∗ |, · · · , |Xq∗ |) denotes the least common multiple of numbers |X1∗ |, · · · , |Xq∗ |. On the other hand, ∑ |Xk | ∏ n 1≤k≤q )q = ( )q . |Xk | ≤ ( q q 1≤k≤q
Obviously, n n max ( )q ≤ exp ( ). q exp
1≤q≤n
Eventually, we arrive at the estimate T ∗ ≤ exp (
n ). exp
Let card(X) = n. A possitive number T ∗ (n) is called a period of the family of all homogeneous dynamical system defined on X if T ∗ (n) is a least possitive natural number such that every homogeneous dynamical system defined on X is periodical with period T ∗ (n). The following assertion is valid. Theorem 8. Let card(X) = n. Let (pk )1≤k≤q be the family of all simple natural numbers such that ∀k)(1 ≤ k ≤ q → pk ≤ n). Let us denote by nk (1 ≤ k ≤ q) a largest natural number such that pnk k ≤ n. Then the following formula T ∗ (n) =
q ∏
pnk k
k=1
is valid. Remark 5. If one assumes the existence of a family X = {xk }1≤k≤n of symbols (letters) such that for some natural number m every word, being in profit, can be represented as the concrete sequence (yi )1≤i≤m , where yi ∈ X 42
for arbitrary i, then the process of development of the native language can be considered as the system (f (x, t))x∈X,t∈Z . The step of time is the mean time which defines the mean leaving time of the native language. The word transits from one form x to another form f (x, t) after time t. It is also assumed that the trajectory of a word x is, as usual, the set of all words (so-called synonyms) (f (x, t))t∈Z which represent only one sense during all time of motion. For example, if x corresponds to the word ”table” at the instant t = 0, then f (x, t) also corresponds to the word ”table” at an arbitrary instant t. Assuming that the system f (x, t)x∈X,t∈Z is dynamical, by Theorem 7 it can be easily concluded that there exists a natural number T ∗ such that the above system is periodical with period T ∗ . This quotation means that each dead language is a living language (or is profit) after concrete time T ∗ . If we consider Theorems 7 and 8 for an analogous genetic model of beings, then in our assumptions each creature is ’repeated’ (in the sense of genetic composition) after the concrete interval of time T ∗∗ (by the unit of time is denoted the mean living time of the family). Theorems 7 and 8 can be used in the information transmission theory which is an important part of mathematical cybernetics. Now we would like to give the well-known applications of Theorems 5 and 6. Example 7. Let us show that the state of the nine planets of the solar system is stable in the sense of Poisson. Indeed, it can be assumed withous loss of generality that each planet is a point with mass 1 and moves in its orbit around the Sun not influenced by the motions of the other planets. Then the phase space of such a system is a 27-dimensional space in R27 , because every planet has three spatial coordinates. Since the k-th planet moves along the circle, its spatial coordinates (xk1 , xk2 , xk3 )(1 ≤ k ≤ 9) satisfy the condition x2k1 + x2k2 + x2k3 = rk2 , where rk is the radius of the k-th planet’s orbit. This means that the configuration point x representing the state of all nine planets moves along the boundary of the 9-dimensional torus, 9 ∏ T9 = Si , i=1
where, for arbitrary i (1 ≤ i ≤ 9) the circle with radius ri is denoted by Si . Let Gk (1 ≤ k ≤ 9) be the group of all rotations of the circle Sk , and let λk be the Haar probability measure defined on B(Sk ). It is clear that, on the one n n n ∏ ∏ ∏ hand, λk is the Gk -invariant probability Borel measure defined on Si . k=1
k=1
k=1
On the other hand, the joint motion of the planet system can be described by the formula f (x, t) = (gω1 (x1 , t), · · · , gω9 (x9 , t)), 43
where gωk (xk , t) is the uniform motion of the k-th planet along the circle Sk with a constant angular velocity ωk . It is clear that, for arbitrary t ∈ R and n n ∏ ∏ x ∈ Gk , the value f (x, t) is an element of the group Gk and, using k=1
k=1
Theorems 5 and 6, we easily conclude that the joint motion of the solar planet system is stable in the sense of Poisson. Remark 6. It is interesting to observe that as far back as 1734 Daniel Bernulli considered the plane of orbits of the planets known at the time as accidental points on the sphere surface and showed that they are uniformly distributed. We must say that the theory of uniform distributions is nowadays well developed (see, for example, [84]). Remark 7. The trajectory of each planet of the solar planet system is an ellipse with the Sun put at one of its focuses. Nevertheless, Theorems 5 and 6 hold for such a model of joint motions of the solar planet system. In some situations, where a metric space X is equipped with a σ-finite invariant measure, one cannot deduce that µ-almost every trajectory is stable in the sense of Poisson. Indeed, let us consider the system of differential equations dxi dx2 dxn = 1, = 0, · · · , =0 dt dt dt whose solution has the form 0)
(0)
x1 = x1 + t, x2 = x2 , · · · , xn = x(0) n and tends to infinity as t → +∞ or t → −∞. It is clear that the solution is not stable in the sense of Poisson. We say that a point y is an ω-limit point for motion (f (x, t))t∈R (x ∈ X) if, for an arbitrary parametric sequence (tn )n∈N tending to +∞, the point y is a limit point for the sequence (f (x, tk ))k∈N . Analogously, a point z ∈ X is called an α-limit point for the motion (f (x, t))t∈R if, for an arbitrary parameter sequence (tk )k∈N tending to −∞, the point z is a limit point for the sequence (f (x, tk ))k∈N . We say that a point x ∈ X tends to ∞ as t → +∞ if the trajectory (f (x, t))t∈R has no ω-limit points. Analogously, a point y ∈ X is called tending to ∞ as t → −∞ if the trajectory (f (y, t))t∈R has no α-limit points. The following proposition is a generalization of the Poincar´e- Carath´eodory theorem. Theorem 9 (E.Hopf ) Let X be a locally compact separable metric space, and let (f (x, t))x∈X,t∈R be a dynamical system with an invariant σ-finite Radon 44
measure. Then µ-almost every point of X is stable in the sense of Poisson or tends to infinity as t → ∞. The proof of Theorem 9 can be found e.g. in [95]. The following important statement was obtained by Birkhoff and Chintchin (see [46]). Theorem 10. Let (E, S, µ) be a measurable space with probability measure and g be some measurable transformation of E. If the measure µ is invariant under the transformation g, then for an arbitrary µ-integrable function f we have n−1 1∑ a) µ({x : (∃ lim f (g k x) = f ∗ (x)}) = 1, n→∞ n k=0
∗
where the function f is µ-integrable and µ-almost invariant under the operator A, i.e. µ({x : x ∈ E & f ∗ (g(x)) = f ∗ (x)}) = 1, ∫ ∫ ∗ b) f (x)dµ(x) = f (x)dµ(x). E
E
If we look at the ergodic sense of a dynamical system, we see that the phenomenon of the existence of an invariant measure in the phase space plays an important role. But it is not always that the phase space is metrically isomorphic to a finite-dimensional Euclidean space. We often have situations in which the phase space of a dynamical system coincides with some-infinite dimensional topological vector space. Hence the problem of developing the ergodic theory in infinite-dimensional topological vector spaces is closely connected with the problem of the existence of invariant and quasiinvariant Borel measures in such spaces. The main goal of our work is to develop the theory of invariant and quasiinvariant measures in some infinite-dimensional topological vector spaces which are phase spaces for a wide class of dynamical systems. As is well-known, non-trivial σ-finite Borel measures, being invariant under the group of all translations, do not always exist in infinite-dimensional topological vector spaces. However, in these spaces it is sometimes possible to construct σ-finite nontrivial Borel measures invariant under some everywhere dense vector subspaces. Such measures defined in infinite-dimensional topological vector spaces and being invariant in the above-mentioned sense will be the object of our interest in the sequel. It should be said that invariant σ-finite Borel measures defined in infinitedimensional topological vector spaces do not have the uniqueness property and, 45
therefore, so one can obtain a wide spectrum of ergodic models for a concrete dynamical system, but in certain situations the uniqueness of the considered measure can be studied by the usual operation of completion. When one wants to construct such a nontrivial translation-invariant Borel measure in the infinite-dimensional topological vector space RN , analogous (in the sense of the invariance under translationes) to the Lebesgue measure in the finite-dimensional Euclidean space Rn , one must require that the value of such measure on infinite-dimensional cube [0; 1[N is equal to 1. This measure cannot be σ-finite, since the space RN can be represented as a continual union of disjoint translates [0; 1[N . The following proposition will be proved in Section 6. Theorem 11. In the consistent formal system of axioms (ZF )&(DC)& (every subset of R is measurable in the Lebesgue sense) there exists a non-trivial translate-invariant Borel measure in RN having the value one on the infinite-dimensional cube [0; 1[N . Remark 8. Using the method described in Section 6, we can construct analogous Borel measures in various infinite-dimensional topological vector spaces. Now let us consider the problem of the existence of invariant and quasiinvariant Borel measures in some infinite-dimensional separable Banach spaces. In this context, the following auxiliary proposition will be helpful (see, e.g. [75]). Lemma 1. An arbitrary infinite-dimensional separable Banach space H is the continuous linear image of the space ℓ1 , where, as usual, ℓ1 = {(xk )k∈N : (xk )k∈N ∈ RN &
∞ ∑
|xk | < ∞}.
k=1
Proof. Let us denote by || · ||H and || · ||ℓ1 the norms in the Banach spaces H and ℓ1 , respectively. Let (xn )n∈N be a sequence of elements in the unit ball B0 ⊂ H, being everywhere dense in B0 , where B0 = {h : ||h||H < 1}. We can put (∀(an )n∈N )((an )n∈N ∈ ℓ1 → Φ((an )n∈N =
∞ ∑ n=1
46
an xn ).
Let us verify that the above definition is correct. To this end, first we have n ∑ to show that the series of elements ( ak xk )n∈N is convergent. For this it is k=1
sufficient to prove that the necessary and sufficient Cauchy condition is valid. Indeed, on the one hand, ∑ (∀ϵ)(ϵ > 0 → (∃nϵ )(n ≥ nϵ → |ak | < ϵ) k≥nϵ
and, on the other hand, ∑ ∑ || ak · xk ||H < ||ak · xk ||H = k≥nϵ
=
∑
k≥nϵ
|ak | · ||xk ||H
0 for all t ∈ T . We can prove that the total mass of the set F ( with respect to the measure WI ) does not appear but lim WI (f (F, t)) = 0.
t→∞
56
Note that the mass disappearance velocity (with respect to the measure WI ) depends on how large the module ||a|| of the velocity a is. On the other hand, we cannot indicate a finite instant t∗ at which the mass fully disappears. It is clear that this phenomenon characterizes linear motions in RI (from the point of view of Gaussian measure) when the cardinality of I is finite. II. Card (I) ≥ ℵ0 . Let F be any Borel subset of U ⊆ RI with WI (F ) > 0, where U is the support of the canonical Gaussian Borel measure WI . It is clear that the velocity of a point (ai )i∈I of RI is equal to a. Two ∑ 2subcases are possible here: a) ai < +∞. i∈I
So a ∈ ℓ2 . By using Theorem 3 we conclude that WI (f (F, t)) > 0 at all time instants. It means that we have situations in which the mass of the set does not appear fully. ∑ 2 b) ai = +∞. i∈I
So a ∈ / ℓ2 (I). We easily conclude that at an arbitrary instant t ̸= 0 we also have at ∈ / ℓ2 . By using Theorem 3, we get the validity of the condition WI (f (F, t)) = 0. The latter relation shows that the ”rest” mass of the set F (with respect to the canonical Gaussian measure) fully appears during the entire motion time in the case of a linear motion in RI when the ”norm” of velocity is infinite. If we look at the inverse motion, then we discover that the opposide situation is possible, i.e., some sets whose ”rest” mass is zero and whose velocity is ”infinite” may acquire positive ”moving” mass (with respect to the canonical Gaussian measure). We can compare the latter proposition to the well-known postulate on which Einstein’s theory is based,namely, this postulate says: If the velocity v of a moving particle is equal to the light velocity c, then the rest mass m0 is equal to zero. Remark 3. Let f (x, t)x∈X,t∈R be some dynamical system, B(X) be some σ-algebra of subsets, invariant under the group of all motions generated by this dynamical system, µ be some nontrivial measure defined on B(X). Usually, the value µ(f (F, t)), where F ∈ B(X) and t ̸= 0, is called the moving mass (with respect to the measure µ) at the instant t. For t = 0 the value µ(f (F, t)) is called the rest mass and it coincides with the value µ(F ). The statement that the measure µ is invariant under the group of all motions means that at an arbitrary instant mass at rest and moving mass become equal. Analogously, the quasiinvariance under the group of all motions means that the sign of mass is preserved during the entire motion time. If we consider the family of various measures defined on B(X), then the relativity law of mass at rest (under the family of measures ) is as usual. 57
Remark 4. Using the method considered in Section 7, one can construct a translation-invariant measure µ defined in RN and taking value 1 on the infinite-dimensional cube [0; 1]∞ . If we consider linear motions in RN from the point of view of measure µ, then the laws of Newtonian classical mechanics (in view of mass) are invariable.
58
4. Construction of Probability Borel Product-Measures in the Topological Vector Space RI by the Methods of Haar Measure Theory Let us recall some definitions from probability theory. Let I be an arbitrary nonempty set of parameters. Denote by RI the vector space of all real functions defined on I. Assume that B(RI ) is the σ-algebra of all Borel subsets of the space RI , generated by the Tikhonov topology τ ; Let (P ri )i∈I be the family of projections defined by (∀i)(∀(xj )j∈I )(i ∈ I & (xj )j∈I ∈ RI → P ri ((xj )j∈I ) = xi ). A minimal σ-algebra of subsets of RI generated by the class of subsets is denoted by Cl(RI ) and is called the cylindrical σ-algebra of subsets of R . Remark 1. Note that Cl(RI ) = B(RI ) for card(I) ≤ ℵ0 . If card(I) > ω, then
(pri−1 (X))i∈I,X∈B(R) I
Cl(RI ) ⊂ B(RI ) & B(RI ) \ Cl(RI ) ̸= ∅. As usual, a measure defined on B(RI ) is called a Borel measure. Analogously, a measure defined on Cl(RI ) is called a cylindrical measure. Denote also R(I) = {(xt )t∈I : (xt )t∈I ∈ RI &card{i|xi ̸= 0} < ℵ0 }, where ℵ0 is the cardinality of the set of all natural numbers. Definition 1. A Borel probability measure µ defined on (RI , τ ) is called Radon if (∀B)(B ∈ B(X) → µ(B) =
sup
µ(K) ).
K⊂B
K is compact in X Definition 2. A family (Ui )i∈I of open subsets in (RI , τ ) is called a general sequence if (∀i1 )(∀i2 )(i1 ∈ I & i2 ∈ I → (∃i3 )(i3 ∈ I → (Ui1 ⊂ Ui3 & Ui2 ⊂ Ui3 ))). Definition 3. A Borel probability measure µ defined on (RI , τ ) is called τ -smooth if, for an arbitrary generalized sequence (Ui )i∈I , the condition µ(
∪ i∈I
Ui ) = sup µ(Ui ) i∈I
is valid. 59
Definition 4. A cylindrical probability measure µ defined on (RI , τ ) is called τ -smooth if, for an arbitrary ∪ generalized sequence (Ui )i∈I of open cylindrical subsets in RI , for which Ui is also a cylindrical subset, the condition i∈I
µ(
∪
Ui ) = supµ(Ui ) i∈I
i∈I
is valid (see e.g. [130]). Definition 5. Let µ1 be a cylindrical measure defined in RI . A Borel measure µ2 defined in RI is called a Borel extension of µ1 if (∀X)(X ∈ Cl(RI ) → µ2 (X) = µ1 (X)). Example 1. Let I be an arbitrary nonempty parametric set, and pi be a Borel probability measure defined ∏ in R for all i ∈ I. If card(I) > ℵ0 , then the pi is defined on the σ-algebra probability product-measure i∈I
∏
B(Ri ) = Cl(RI ).
i∈I
Accordingly, this measure is an example of a cylindrical probability measure which is not defined on B(RI ). The following definition from general topology is important for our investigation. Definition 5. Assume that for arbitrary i ∈ I, fi is a mapping of the topological∏space (Xi (1) , τi (1) ) into the topological space (Xi (2) , τi (2) ). Then the fi defined by mapping i∈I
(∀(xi )i∈I )((xi )i∈I ∈
∏
Xi (1) → (
i∈I
∏
fi ) ((xi )i∈I ) = (fi (xi ))i∈I )
i∈I
is called the direct product of the family (fi )i∈I (see e.g. [25]). In the sequel, we will need some auxiliary results. Lemma 1. If a cylindrical measure P defined on a topological space (RI , τ ) is τ -smooth, then, in (RI , τ ), there exists only one τ -smooth Borel extension of P. For the proof of Lemma 1, in the general case see e.g. [130], p.39. 60
Lemma 2. The direct product
∏
fi of the family (fi )i∈I is continuous if
i∈I
and only if (∀i)(i ∈ I → fi is continuous). The proof of Lemma 2 is easy and is given e.g.in [25]. Remark 2. It is easy to verify that if ∏
∏
fi is a continuous mapping, then
i∈I
fi is also a Borel measurable mapping.
i∈I
Remark 3. If for an arbitrary i ∈ I a mapping fi is a (B(Xi (2) ), B(Xi (1) ))∏ ∏ ∏ measurable, then fi is a (CL( Xi (2) ), CL( Xi (1) ))-measurable mapping, i∈I
i∈I
i∈I
where B(Xi (k) ) denotes the σ-algebra of subsets of Xi (k) generated by the topol∏ ogy τi (k) (1 ≤ k ≤ 2&k ∈ N ); by CL( Xi (k) ) is denoted the cylindrical σi∈I
algebra of subsets of the product topological space (
∏ ∏ Xi (k) , τi (k) )(1 ≤ k ≤ 2 & k ∈ N ). i∈I
i∈I
Lemma 3. Let, for an arbitrary i ∈ I, pi be a probability Borel measure defined on R. Then there exists a family (fi )i∈I of Borel measurable real functions defined on ]0; 1[ such that (∀i)(∀y)(i ∈ I & y ∈ R → l1 ({x|x ∈]0; 1[ & fi (x) ≤ y}) = pi ((−∞; y]) ), where l1 denotes the Lebesgue measure on ]0; 1[. This lemma is well known and its proof can be found e.g.in [115], Example 3.25. Lemma 4. Let (E1 , τ1 ) and (E2 , τ2 ) be two topological spaces. Denote by B(E1 ) and B(E2 ) (correspondingly, B(E1 × E2 )) the class of all Borel subsets generated by the topologies τ1 and τ2 (correspondingly, τ1 × τ2 ). If at least one of these topological spaces has a countable base, then the equality B(E1 ) × B(E2 ) = B(E1 × E2 ) holds.
61
Proof. Let E1 have a countable base of open sets. Denote by (Bn )n∈N some base of this space. Lemma 4 will be proved if we show that an arbitrary open set in E1 × E2 can be expressed by the union ∪n∈N An , where, for an arbitrary n ∈ N , An is an elementary open set in E1 × E2 . The latter means that, for An , we have the representation An = An (1) × An (2) , where An (k) is an open set in Ek (1 ≤ k ≤ 2). Obviously, we can write G = ∪t∈I Ut for an arbitrary open set G in E1 × E2 , where a) I is some set of parameters; b) (∀t)(∀k)(t ∈ I & (1 ≤ k ≤ 2) → (Ut (k) is an open set in E (k) )) . For an arbitrary parameter t ∈ I denote by θt the set of all natural numbers for which we have ˜n (t) = ∪n∈θ Bn × Ut (2) . U t Let us put { ˜t = U
(2)
(2)
Ut , if Bn × Ut ⊂ G, (2) ∅, if Bn × Ut ̸⊂ G.
On the one hand, if x ∈ G = ∪t∈I Ut , then there exist t0 ∈ I and n0 ∈ θt0 such that x ∈ Bn0 × Ut0 (2) . This means that (2)
˜n (t0 ) ⊂ ∪n∈N (Bn × ∪t∈I U ˜n (t)). Bn0 × Ut0 = Bn0 × U 0 On the other hand, if ˜ (t), x ∈ ∪n∈N (Bn × ∪t∈I U then there exist n0 ∈ N and t0 ∈ I such that ˜n (t0 ). x ∈ Bn0 × U 0 . ˜n (t) we have By the definition of the set U 0 Bn0 × Ut0 (2) ⊂ G. 62
This completes the proof of the lemma. Denote by Si the unit circle in the Euclidean space R2 . We can identify the circle Si with a compact group of all rotations of R2 . Lemma 5. Assume that λI is a probability Haar measure defined on the ∏ group Si . Then, for card(I) > ℵ0 , the set X defined by i∈I
X=
∏ (Si \ {(0; 1)i }) i∈I
is a λI -massive nonmeasurable subset of the group
∏
Si .
i∈I
Proof. Let us show that the inner λI -measure of the set X is equal to zero. If we assume the contrary, then by using the inner regularity of the Haar measure (see [38]) we obtain the existence of a compact subset F ⊂ X such that λI (F ) > 0. It is clear that λi (P ri (F )) < 1, where λi is the probability Haar ∏ Sj onto Si . measure on Si and P ri is the projection from j∈I
∏ λI ( P ri (F )) = 0
We have
i∈I
because Card(I) > ℵ0 . This contradicts the conditions ∏ F ⊂ P ri (F ), λI (F ) > 0. i∈I
As the Haar measure λ is τ -smooth, we obviosly conclude that ∏ λ∗I ( (Si \ {(0.1)i })) = 1. i∈I
This finishes the proof. Remark 4. The results of Lemmas 4 and 5 were obtained in [70]. Define the measure λ(I) by (∀B)(B ∈
∏ i∈I
∏ Si → λ(I) ( (Si \ {(0; 1)i } ∩ B)) = λI (B)). i∈I
. ˜ (I) the completion of the measure λ(I) and define µI by Denote by λ ∏ ˜ (I) ) → µI (( gi )−1 (X)) = λ ˜ I (X)), (∀X)(X ∈ dom (λ i∈I
63
where 3 3 (∀x)(x ∈]0; 1[ → gi (x) = (cos(2πx − π), sin(2πx − π)) ). 2 2 ˜ Denote by B(]0; 1[I ) the σ -algebra of µI -measurable subsets of ]0; 1[I . The following result is of some interest. Theorem 1. Let (pi )i∈I be an arbitrary family of Borel probability measures defined on R. Let (fi )i∈I be the family of Borel measurable functions constructed ∏ ˜ in Lemma 3. If the mapping fi is a (B(RI ), B(]0; 1[I ))-measurable mapi∈I
ping, then in RI there exists a Borel extension PI of the cylindrical probability ∏ product-measure pi . i∈I
Proof. Define the functional PI by
∏ (∀X)(X ∈ B(RI ) → PI (X) = µI (( fi )−1 (X))). i∈I
Show that the measure PI is an extension of the product-measure Indeed, for every finite parameter set I0 ⊂ I and for a cylindrical set ∏ (−∞; xi ] (i ∈ I0 , xi ∈ R)
∏
pi .
i∈I
i∈I0
we have PI (RI\I0 ×
∏
∏ ∏ (−∞; xi ]) = µI (( fi )−1 (RI\I0 × (−∞; xi ]) =
i∈I0
i∈I
= µI (]0; 1[I\I0 ×
∏
i∈I0
{y : y ∈ ]0; 1[ & fi (y) ≤ xi }).
i∈I0
In view of Lemmas 3 and 4, using the property of a Haar measure, we have ∏ µI (]0; 1[I\I0 × {y : y ∈]0; 1[ & fi (y) ≤ xi }) = i∈I0
= µI\I0 (]0; 1[I\I0 ) × µI0 ( =
∏ i∈I0
∏
{y : y ∈]0; 1[ & fi (y) ≤ xi }) =
i∈I0
l1 ({y : y ∈]0; 1] & fi (y) ≤ xi }) =
∏
Fi (x) =
i∈I0
This completes the proof of the theorem. 64
∏ i∈I
pi (RI\I0 ×
∏
i∈I0
(−∞; xi ]).
Remark 5. From the results of Theorem 1 and Lemmas 1–4 we conclude that for an arbitrary family (pi )i∈I of Borel probability measures defined on R and having continuous positive density functions (fi )i∈I , there exists ∏ only one Borel extension PI of the probability cylindrical product-measure pi . In i∈I
particular, if, for i ∈ I, pi is the probability Borel measure defined on R and I having strictly monotone distribution function defined ∏ on R, then in R there exists only one Borel extension PI of the measure pi . i∈I ∏ Remark 6. The mapping fi is always (CL(RI ), CL(]0; 1[I ))-measurable i∈I
and ∏ so we have the following representation of the cylindrical product measure pi constructed by Anderson: i∈I
∏ ∏ (∀B)(B ∈ CL(RI ) → ( pi )(B) = µI (( fi )−1 (B)). i∈I
i∈I
An example of a Borel extension of a concrete cylindrical probability productmeasure defined on RI and quasiinvariant with respect to the vector subspace R(I) was first constructed in [70]. Hence it is of interest to point out, within the class of all Borel extensions of the probability product-measures defined on RI , a subclass of measures whose every element possesses the above-mentioned properties. We will need the following lemma. Lemma 6. Let I be some nonempty set of parameters and, for an arbitrary parameter i ∈ I, let Si be the unit circle in the Euclidean space R2 , and let (pi )i∈I be a family of Borel probability measures whose every element pi is defined on Ri = R and has the distribution function Fi with a continuous positive derivative. For i ∈ I denote by fi the mapping of Ri into Si \ {(0; 1)i } defined by 3 3 (∀x)(x ∈ Ri → fi (x) = (cos(2πFi (x) − π), sin(2πFi (x) − π))). 2 2 If h is an arbitrary translation of the real axis Ri , then it is clear that the mapping fi ◦ h ◦ fi−1 is a continuous automorphism of the space Si \ {(0; 1)i }. This automorphism is uniquely extended to a continuous automorphism of the unit circle. Denote by Gi the group of all such automorphisms of the unit circle. ∏ Then the probability Haar measure λI defined on Si is quasiinvariant with i∈I ∑ respect to the direct sum Gi of the family of groups (Gi )i∈I . i∈I
Proof. Denote by λi the probability Haar measure defined on the unit circle Si (i ∈ I). We will prove the lemma in two steps. 65
1) The measure λi is Gi -quasiinvariant. Note that, for an arbitrary neighbourhood U of the point (0; 1)i and for an arbitrary element g ∈ Gi , there exists a positive real number K(U, g, i) such that (∀x1 )(∀x2 )((x1 ∈ Si \ U ) & (x2 ∈ Si \ U ) → λi ((g(x1 ), g(x2 )) ≤ K(U, g, i) × λi ((x1 , x2 )), where (x1 , x2 ) and (g(x1 ), g(x2 )) denote the open arcs of the unit circle Si which do not contain the point (0; 1)i . Indeed, without loss of generality we can consider a neighbourhood U of the point (0; 1)i having the form (∃α0 )(0 < α0 < π → U = {(x; y)|(x; y) = (cosϕ, sinϕ),
π π − α0 < ϕ < + α0 }. 2 2
Let x1 ∈ Si \ U and x2 ∈ Si \ U . Then for g ∈ Gi there exists a translation h of Ri such that g = fi ◦ h ◦ fi−1 . We have
t∫ 2 +h
Fi′ (t)d(t) = (t2 − t1 ) × Fi′ (ξ1 ) =
λ(g(x1 , x2 )) = t1 +h
F ′ (ξ1 ) = (t2 − t1 ) × i′ × Fi′ (ξ2 ) ≤ (t2 − t1 ) × Fi (ξ2 )
sup Fi′ (t) t∈[t1 +h;t2 +h] inf Fi′ (t) t∈[t1 ;t2 ]
× Fi′ (ξ2 ) ≤
≤ K(U, g, i) × λi ((x1 , x2 )), where x1 = fi (t1 ), x2 = fi (t2 ) (t1 < t2 ) ; t1 + h < ξ1 < t2 + h, t1 < ξ2 < t2 , ∫t2
Fi′ (x)dx = (t2 − t1 ) × F ′ (ξ2 ),
t1
sup K(U, g, i) =
t∈[fi−1 (t01 )−|h|,fi−1 (t02 )+|h|]
inf
t∈fi−1 (Si \U )
Fi′ (t)
Fi′ (t) ,
π π π π − α0 ), sin( − α0 )), fi (t02 ) = (cos( + α0 ), sin( + α0 )). 2 2 2 2 Let X be an arbitrary λi -measure zero subset of the space Si . Let (Un )n∈N be a fundamental system of neighbourhoods of the point (0; 1)i . fi (t01 ) = (cos(
66
For an arbitrary element g ∈ Gi , consider the set g(X). We must prove that the set g(X) is a λi -measure zero set. Let us put Xn = (Si \ Un ) ∩ X (n ∈ N ). The relation { g(X) =
∪n∈N g(Xn ) ∪ {(0; 1)i }, if (0; 1)i ∈ X, ∪n∈N g(Xn ), if (0; 1)i ∈ / X.
holds. It is sufficient to show that (∀n)(n ∈ N → λi (g(Xn )) = 0). Let ϵ be an arbitrary positive number. Then by the property of a Haar measure (see e.g.[26]), there exists, in Si , an open set G such that Xn ⊂ G ⊂ Si \ Un , λi (G \ Xn )
0. Withous loss of generality we may assume that λI (B1 \ B2 ) > 0. 68
By the property of the massive set
∏
(Si \ {(0; 1)i }), we have
i∈I
(B1 \ B2 ) ∩ (
∏ (Si \ {(0; 1)i })) ̸= ∅, i∈I
which is a contradiction, and therefore the correctness of the definition of the measure µI is proved. ∑ It is clear that the measure µI is Gi -quasiinvariant. i∈I
Denote by PI the functional defined by
∏ (∀B)(B ∈ B(RI ) → PI (B) = µI (( fi )(B)), i∈I
where (fi )i∈I is the family of mappings constructed in Lemma 5. If B is an arbitrary PI -measure zero subset of the space RI , then we have ∑ ∏ (∀h)(h ∈ Ri → PI (h(B)) = µI (( fi )(h(B))) = i∈I
i∈I
∏ ∏ ∏ ∏ = µI (( fi ) ◦ h−1 ◦ ( fi )−1 ◦ ( fi ) ◦ h(B)) = µI (( fi )(B)) = PI (B)). i∈I
i∈I
i∈I
i∈I
Note that for every finite parameter set I0 and for every cylindrical set ∏ (−∞; xi ] (xi ∈ Ri , i ∈ I0 ) i∈I0
we have PI (RI\I0 ×
∏
∏ ∏ (−∞; xi ]) = µI (( fi )(RI\I0 × (−∞; xi ]) =
i∈I0
∏
= µI ( = µI (
i∈I
(Sj \ {(0; 1)j })) × (
j∈I\I0
∏
(Sj \{(0; 1)i })×
j∈I\I0
= λI\I0 (
S i ) × λ I0 (
Fi′ (x)dx =
i∈I0−∞
fi )(
∏
(−∞; xi ])) =
∏
Sj ×
fi ((−∞; xi ])) = λI0 (
∏
fi ((−∞; xi ])) =
i∈I0
j∈I\I0
i∈I0
∏
∏
i∈I0
fi ((−∞; xi ])) = λI (
i∈I0
∏
xi ∏ ∫
∏
i∈I0
∏
i∈I\I0
=
i∈I0
∏
fi ((−∞; xi ])) =
i∈I0
Fi (xi ) =
i∈I0
∏ i∈I
pi (RI\I0 ×
∏
(−∞; xi ]).
i∈I0
This means that ∏ the measure PI is a Borel extension of the cylindrical product measure pi . i∈I
69
By the property of τ0 -smoothness ∏ of the Haar measure λI (see e.g. [92]), where τ0 denotes the topology on Si , we conclude that the measure µI is i∈I ∏ τ1 -smooth, where τ1 denotes the induced (by τ0 )topology on the space (Si \ i∈I
{(0; 1)i }) . Analogously, by the property of τ1 -smoothness of the probability measure µI and by the equality PI = (
∏
fi )−1 ◦ µI ,
i∈I
we conclude that the measure PI is τ -smooth, where τ denotes the Tikhonov’s topology in the space RI . ∏ Finally, by the property of τ -smoothness of PI , the product measure pi i∈I
is also τ -smooth and, using the result of ∏Theorem 1, we conclude that the Borel extension PI of the product measure pi is unique. i∈I
The proof is comleted. Let us consider some corollaries of Theorem 2. Corollary 1. The main result of [70] can be obtained if we assume that every element of the above-mentioned family is a Cauchy probability measure, i.e. 1 1 x (∀i)(∀x)(i ∈ I&x ∈ R → Fi (x) = + · arctg( )). 2 π 2 Corollary 2. The product of an arbitrary family (pi )i∈I of nontrivial Gaussian Borel probability measures defined on RI has only one Borel extension which is quasiinvariant with respect to the vector subspace R(I) . The property of τ -smoothness of the Gaussian cylindrical measure in RI is easily∏proved using Si . (cf. the the property of τ - smoothness of the Haar measure λI defined on i∈I
proof of the general result obtained in [128]). Corollary 3. In the case of the space RI , for Card(I) > ℵ0 Theorem 5 is a generalization of the Kakutani theorem which gives only the construction of quasiinvariant cylindrical measures. Note that, for Card(I) > ℵ0 , in the vector space RI there exists no Radon probability measure which would be quasiinvariant with respect to the vector subspace R(I) (see section 5, also [102]). For Card(I) > 2ℵ0 in the vector space RI there exists no Radon probability measure which would be quasiinvariant with respect to an everywhere dense subspace of RI (see [102], also Section 5 below). 70
If Card(I) = 2ℵ0 , then in the vector space RI there exists a nontrivial σfinite Radon measure which is invariant with respect to some everywhere dense subspace of RI (see [102]). We must say that the following two problems have not so far been solved. Problem 1. Does there exist a nontrivial σ-finite Borel measure in the space RI for Card(I) ≥ ℵ1 which would be invariant with respect to the vector subspace R(I) of the space RI ? Problem 2. Does there exist a nontrivial σ-finite Borel measure in the space RI for Card(I) > 2ℵ0 which would be invariant with respect to some everywhere dense vector subspace of RI ? In connection with these problem, see also more general problems posed in [72]. Let I be an arbitrary infinite parameter set, and µ be a canonical probability Gaussian Borel measure defined in RI . The following natural problem arises: derive a characterization of the vector space of all admissible translations of the Borel probability product-measures defined in RI . A solution of this problem is given below. Let X =
∏
Xk be the product of countable measurable spaces, and let µk
k∈N
and νk (k ∈ N ) be probability measures such that: 1) µk is absolutely continuous with respect to νk , k (x) 2) dµ dνk (x) = ρk (x). ∏ ∏ νk . µk and ν = Let us consider the product-measures µ = k∈N
k∈N
The following important statement due to Kakutani is valid. Theorem 4 (Kakutani). The measures µ and ν are equivalent ∫ √ if and only ∏ αk is divergent to zero, where αk = if an infinite product ρk (xk )dνk (xk ). k∈N n ∏
In this case rn (x) = r(x) =
∞ ∏
Xk
ρk (x) is convergent (in the mean) to the function
k=1
ρk (x) which is the density of the measure µ with respect to ν, i.e.,
k=1
r(x) =
Proof. Let us show that if
∏
dµ(x) . dν(x)
αk is convergent to zero, then the measures
k∈N
71
µ and ν are orthogonal. So far as ∫ √ ∫ 2 2 αk = | ρk (xk )dνk (xk )| ≤ ρk (xk )dν(xk ) = 1, Xk
the product
∏
Xk
αk cannot be convergent to infinity. If this product is divergent
k∈N
m ∏s
to zero, then there exists a sequence βs =
αk such that the series
∞ ∑
βs is
s=1
k=ns
convergent. Let us consider the sequence of cylindrical sets As = {x :
ms ∏
ρk (xk ) ≥ 1}.
k=ns
Note that v ∫ u∏ u ms dν(x) ≤ t ρk (xk )dν(x) =
∫ ν(As ) = As
ms ∫ ∏ √
=
k=ns
As
ρk (xk )dνk (xk ) = βs .
k=nsX
k
From the convergence of the series
∞ ∑
βs we have
s=1
ν(A) = 0, where A = lims→∞ As . On the other hand, if Bs = X \ As , then ∫
∫ dµ(x) ≤
µ(Bs ) = Bs
{ Bs
ms ∏
ρk (xk )}− 2 dµ(x) = 1
k=ns
v ∫ u∏ ms ∫ ∏ √ u ms t = ρ(xk )dν(x) ≤ ρk (xk )dνk (xk ) = βs . Bs
k=ns
k=nsX
k
Hence µ(lims→∞ Bs ) = 0. The latter relation implies µ(A) ≥ µ(limAs ) = 1, i.e., µ(A) = 1 and we conclude that µ ⊥ ν. 72
∏
Now, assume that the product
αk is convergent to zero. Let us consider
k∈N
the sequence of functions (Φn )n∈N , where v u n u∏ ρk (xk ) (n ∈ N ). Φn (x) = t k=1
From the validity of the relation ∫ |Φn+p (x) − Φn (x)|2 dν(x) =
∫ ∏ n
v u n+p u ∏ ρk (xk )|t ρk (xk ) − 1|2 dν(x) =
X k=1
X
∫
∫ ···
= Xn+1
k=n+1
v u n+p n+p n+p ∏ ∏ u ∏ ρk (xk ) − 1|2 dνk (xk ) = 2(1 − αk ), |t k=n+1
Xn+p
k=n+1
k=n+1
we easily deduce that Φn (x) is a fundamental sequence in L2 (X, ν). So far as ∫ ∫ 1 |rn+p (x) − rn (x)|dν(x) ≤ { |Φn+p (x) − Φn (x)|2 dν(x)} 2 × X
X
∫ 1 ×{ [|Φn+p (x)| + |Φn (x)|]2 dν(x)} 2 ≤ X
∫ 1 ≤ 2{ |Φn+p (x) − Φn (x)|2 dν(x)} 2 , X
Φ2n
the sequence rn = is convergent in the mean. Let r(x) = lim rn (x). For an arbitrary bounded cylindrical function f , the n→∞ equalities ∫
∫ f (x1 , · · · , xn )
f (x)dµ(x) = lim
n→∞
X
f (x1 , · · · , xk )rn (x)
= lim
n→∞
n ∏
∫ dνk (x) = lim
∫ = lim
∫ f (x)rn (x)dν(x) =
n→∞
f (x)rn (x)dν(x) =
n→∞
k=1
X
dµk (xk ) =
k=1
X
∫
n ∏
X
f (x)r(x)dν(x) X
are valid. 73
X
get
If we approximate any measurable function by cylindrical functions, then we ∫ ∫ f (x)dν(x) = f (x)r(x)dν(x). X
X
and the theorem is proved. Corollary 5.(A.V.Skhorokhod Let, for an arbitrary k ∈ N , µk be the canonical Gaussian probability Borel measure ∏ defined in R and having mean 0 and variance 1. Let X = RN and µ = µk be a canonical Gaussian Borel k∈N
probability measure in RN . Then, since the measure µ is quasiinvariant, the vector space Mν of all admissible translations coincides with ℓ2 . Proof. Let us denote by µa the measure defined by (∀B)(B ∈ B(RN ) → µa (B) = µ(B + a)), where a = (ak )k∈N ∈ RN . Obviously, we have
∏
µa =
νk ,
k∈N
∫
where (∀B)(B ∈ B(R) → νk (B) =
(tk −ak )2 1 √ e− 2 dt). 2π
B
Indeed, we get ∫ √ t2 (tk −ak )2 (tk −ak )2 1 k √ e− 2 αk = e− 2 + 2 dtk = 2π R
∫ =
a
a2 a2 (tk − k )2 1 2 k − 8k 2 √ e− dtk = e− 8 . 2π
R
It is not difficult to check that ∏
∑
a2 k
k∈N − 8
αk = e
k∈N
is divergent to zero if and only if the series ∑ a2k k∈N
is convergent in the usual sense , i.e., (ak )k∈N ∈ ℓ2 . 74
Corollary 6. Let νk and µk be two Gaussian probability measures with the density functions (x −γ )2 − k 2k dνk (xk ) 1 2σ k = √ e dxk σk 2π and − dµk (xk ) 1 = √ e dxk λk 2π
(xk −bk )2 2λ2 k
,
respectively, for an arbitrary k ∈ N . Let us put X = R∞ . In this case ρk (xk ) =
σk − 2σ21λ2 [(xk −βk )2 σk2 −(xk −γk )2 λ2k ] e k k . λk
Simple calculations show us that αk = √ where A=
1 · C · eD , σ k λk A
√ √ (βk − γk )2 σk2 + λ2k , C = 2σk · λk , D = − 2 . σk + λ2k
Eventually, we get √ αk =
(β −γ )2
− k2 k 2 2σk λk · e 4(σk +λk ) . 2 2 σk + λ k
∏ νk By the Kakutani theorem we conclude that the product-measures ν = k∈N ∏ and µ = µk are equivalent if and only if the infinite product k∈N
∏ k∈N
√
(β −γ )2
− k2 k 2 2σk λk · e 4(σk +λk ) 2 2 σk + λ k
is divergent to zero. Corollary 7. Let µ be the canonical Gaussian measure defined in RN . Let U : RN → RN be a transformation of RN into itself defined by U ((xi )i∈N ) = (λi xi )i∈I , 75
where (xi )i∈N ∈ RN and λi > 0. Then, by Corollary 6, the Gaussian measure µ is quasiinvariant with respect to U if and only if the product √ ∏ 2λi 1 + λ2i i∈N
is divergent to zero. Corollary 8. Let V be a transformation defined by V ((xk )k∈N ) = (λk xk )k∈N , where λk = (1 +
√
1 − e2βk )e−βk ,
(∀k)(k ∈ N → βk ≤ 0) & (βk )k∈N ∈ ℓ1 . It is clear that
∞ ∏ k=1
√
∑ ∞ βk ∏ 2λk βk k∈N = e = e 1 + λ2k k=1
and, by Corollary 7, the measure µ is quasiinvariant with respect to V . The following result is valid. Theorem 5. Let α be an arbitrary infinite parameter set, and µ be the canonical Gaussian Borel measure defined in Rα . Then, in view of the quasiinvariance of the measure µ, the vector space Mµ of all admissible translations coincides with ℓ2 (α), where ∑ ℓ2 (α) = {(xi )i∈α ∈ Rα & x2i < ∞}. i∈α
Proof. Let us denote by µJ the canonical Gaussian propability measure defined on RJ (J ⊆ α). If we denote by µα\J the canonical Gaussian Borel probability measure defined on Rα\J , then, using Lemma 4, we get (∀J)(J ⊆ α & Card(J) ≤ ℵ0 → µ = µJ × µα\J ). Ad hoc µJ is an ℓ2 -quasiinvariant measure, µ is ℓ2 (J) × Iα\J quasiinvariant for all J ⊆ α with card(J) = ℵ0 , where Iα\J is the identity mapping of Rα\J onto itself, which can be identifed with the degenerate translation of Rα\J . It is clear that ℓ2 (α) ⊆ ∪J⊆α,card(J)=ℵ0 ℓ2 (J) × I α\J . 76
Let (xi )i∈α ∈ Mµ . Let us assume the contrary and let (xi )i∈α ∈ / ℓ2 (α). Then ∑ x2i = +∞. i∈α
It is clear that we can indicate a countable subspace J0 = (im )m∈N ⊆ α such that ∑ x2i = +∞. i∈J0
By Corollary 5, we can choose a Borel set B0 ⊆ RJ0 such that µJ0 (B0 ) > 0 & µJ0 (B0 + (xi )i∈J0 ) = 0. Then from the relations µ(B0 × Rα\J0 ) = µJ0 (B0 ) > 0 & µ(B0 × Rα\J0 + (xi )i∈α ) = µ((B0 + (xi )i∈J0 ) × (Rα\J0 + (xi )i∈α\J0 ) = = µ((B0 + (xi )i∈J0 ) × Rα\J0 ) = µJ0 (B0 + (xi )i∈J0 ) = 0 we get a contradiction to the condition (xi )i∈I ∈ Mµ , and Theorem 5 is proved. Note that using Theorems 3, 4 and Lemma 4, we can characterize the vector space of all admissible translations for probability product-measures defined in Rα and being the product of absolutely continuous (with respect to the Lebesgue measure) probability measures (pi )i∈α . Indeed, let ai ∈ Ri (i ∈ α). Denote by (a ) pi i the measure defined by (ai )
(∀B)(B ∈ B(R) → pi
(B) = pi (B + ai )).
We put (a )
(ai )
ρi If we write αi =
=
dpi i . dpi
∫ √ (a ) ρi i (xi )dpi (xi ), R
then, for Mν , we get the representation Mν = {(ai )i∈α : (∀J)(J ⊆ α & card(J) ≤ ℵ0 →
∏ i∈J
77
αi is divergent to zero},
where ν =
∏
pi .
i∈N
A characterization of all transformations of RI generated by infinite diagonal matrices under which canonical Gaussian Borel probability measures are quasiinvariant, is presented by the following theorem. Theorem 6. Let α be an arbitrary set of parameters. Then the canonical Gaussian probability Borel measure µ defined in Rα is quasiinvariant with respect to U having the form U ((xi )i∈α ) = (λi xi )i∈α if and only if there exists α0 ⊆ α with card (α0 ) ≤ ℵ0 such that the product √ ∏ 2λi 1 + λ2i i∈α 0
is divergent to zero and (∀i)(i ∈ α \ α0 → λi = 1).
Proof. Sufficiency. Assume that the sufficient condition holds. Denote by U : Rα0 → Rα0 the transformation defined by U ((xi )i∈α0 ) = (λi xi )i∈α0 . Analogously, denote by V : Rα\α0 → Rα\α0 the identity transformation of Rα\α0 into itself defined by V ((xi )i∈α\α0 ) = (xi )i∈α\α0 . Denote by µα0 and µα\α0 the canonical Gaussian measures defined on Rα and Rα\α0 , respectively. It is clear that µ = µα0 × µα\α0 . From Corollary 7 it follows that µα0 is quasiinvariant under the transformation U . The measure µα\α0 can be considered as quasiinvariant under the 78
transformation V . Hence the measure µ is quasiinvariant under the transformation U × V and the sufficientcy is proved. Necessity. Let µ be quasiinvariant under the transformation U . It is clear that for an arbitrary countable parameter set α0 ⊆ α, the product √ ∏ 2λi 1 + λ2i i∈α 0
is divergent to zero. ∏ √ 2λi Indeed, if we assume that there exists α0 ⊆ α such that is 1+λ2 i∈α0
i
convergent to zero, then by Corollary 7 the canonical Gaussian Borel measure µα0 is not quasiinvariant with respect to U : Rα0 → Rℵ0 , where U ((xi )i∈α0 ) = (λi xi )i∈α0 . The latter relation means that there exists B0 ∈ B(Rℵ0 ) such that µα0 (B0 ) > 0 & µα0 (U (B0 )) = 0. Let us define a transformation V : Rα\α0 → Rα\α0 by V ((xi )i∈α\α0 ) = (λi xi )i∈α\α0 . It is clear that µ(B0 × Rα\α0 ) = µα0 (B0 ) × µα\α0 (Rα\α0 ). On the other hand, we have µ(U (B0 × Rα\α0 )) = µα0 (U (B0 ) × V (Rα\α0 )) = = µα0 (U (B0 )) × µα\α0 (V (Rα\α0 )) = = µα0 (U (B0 )) × µα\α0 (Rα\α0 ) = µα0 (U (B0 )) = 0 and we get a contradiction with the condition of quasiinvariance of the measure µ under the transformation U . Let us assume that card({i : λi ̸= 1}) > ℵ0 . Obviously, we get √ card({i : 0
ℵ0 . Then in the measurable space (RI , B(RI )) there exists no Radon probability measure which would be quasiinvariant with respect to the vector space R(I) . The proof of this theorem can be obtained by assuming the contrary. Then we obtain a contradiction to the property of σ-finiteness of the measure. In connection with Example 2, the following result is also of interest. Theorem 4. If I is a set of indices with card(I) > c (the cardinality of the continuum is denoted by c), then in the measurable space (RI , B(RI )) there exists no Radon probability measure which would be quasiinvariant with respect to some everywhere dense vector subspace of RI . Let us recall one important definition. Definition 3. Assume that Γ is a family of real functions defined on the set X. A family Γ is called separating the points of the set X if (∀x1 )(∀x2 )(x1 ∈ X & x2 ∈ X & x1 ̸= x2 → (∃f )(f ∈ Γ → f (x1 ) ̸= f (x2 ))). Lemma 3. If card(X) > c, then an arbitrary countable family Γ of real functions defined on the set X does not separate the points of the set X. Proof. Indeed, to an arbitrary point x ∈ X let us put into the correspondence a point y(x) ∈ RN such that y(x) = (f1 (x), f2 (x), · · ·), where Γ = {fk }k∈N . Since card(RN ) = c, there exists a subset Y ⊆ X such that (a) ard(Y ) = card(X). (b) (∀z)(∀x)(z ∈ Y & x ∈ Y → y(x) = y(z)). If x1 ∈ Y , x2 ∈ Y and x1 ̸= x2 , then there is no function f ∈ Γ such that f (x1 ) ̸= f (x2 ) . Lemma 3 is proved. Proof of Theorem 4. 83
Assume the contrary and let µ be a Radon probability measure in the measurable space (RI , B(RI )) which is quasiinvariant with respect to some everywhere dense in RI vector subspace G ⊆ RI . This means that, for some compact subset K of RI , we have 0 < µ(K) < +∞. Then there exists a∪countable family (gn )n∈N of elements of the group G such that n∈N gn (K) is a µ-almost G-invariant subset of the space RI , i.e., ∪ ∪ (∀f )(f ∈ G → µ(( gn (K))△f (( gn (K))) = 0). n∈N
n∈N
Using Lemma 1, we obtain that the family (gn )n∈N does not separate the set I. Therefore there exist two different points i1 ∈ I and i2 ∈ I such that (∀n)(n ∈ N → gn (i1 ) = gn (i2 )). Let us put ( ) M = diam Pri1 (K) ∪ {0} . Applying the density of G, let us consider an element f ∗ ∈ G such that f ∗ (i1 ) ∈]g1 (i1 ) + 2M ; g1 (i1 ) + 3M [. Clearly, ( ) ∪ ∪ µ f ∗( gn (K)) \ gn (K) > 0, n∈N
n∈N
which contradicts the condition (∀f )(f ∈ G → µ(f (
∪
gn (K))△
n∈N
∪ n∈N
Thus, Theorem 4 is proved.
84
gn (K)) = 0).
6. Invariant Borel Measures in the Topological Vector Space RN Let Rn be the space of all sequences of real numbers equipped with the Tikhonov topology. Let us denote by B(RN ) the σ-algebra of all Borel subsets in RN . Let (ai )i∈N and (bi )i∈N be sequences of real numbers such that (∀i)(i ∈ N → ai < bi ). We put An = R0 × · · · × Rn × (
∏
∆i ) (n ∈ N ),
i>n
where (∀i)(i ∈ N → Ri = R & ∆i = [ai ; bi ]). For an arbitrary natural number i ∈ N , consider the Lebesgue measure µi defined on the space Ri and satisfying the condition µi (∆i ) = 1. Let us denote by λi the normed Lebesgue measure defined on the interval ∆i . For an arbitrary n ∈ N , let us denote by νn the measure defined by ∏ ∏ νn = µi × λi , 1≤i≤n
i>n
and by ν n the Borel measure in the space RN defined by (∀X)(X ∈ B(RN ) → ν n (X) = νn (X ∩ An )). The following assertion is valid. Lemma 1. For an arbitrary Borel set X ⊆ RN there exists a limit ν∆ (X) = lim ν n (X). n→∞
Moreover, the functional ν∆ is a nontrivial σ-finite measure defined on the Borel σ-algebra B(RN ). Proof. First, observe that,for an arbitrary natural number n, the condition An ⊂ An+1 is valid. By the property of σ-additivity of the measure νn+1 , we obtain ν n+1 (X) = νn+1 (X ∩ An+1 ) = νn+1 (X ∩ [An+1 \ An ] ∪ An ) = = νn+1 [X ∩ (An+1 \ An )] + νn+1 (X ∩ An ). Note that the restriction νn+1 |An of the measure νn+1 to the set An coincides with the measure νn . 85
Indeed, we have νn+1 (An ∩ X) = (
∏
∏
λi )(An ∩ X) =
i>n+1
1≤i≤n+1
{
∏
µi ×
λi }(An ∩ X) =
i>n+1
1≤i≤n
=(
∏
µi × [µn+1 |∆n+1 + µn+1 |{R \ ∆n+1 }] × ∏
µi ×
1≤i≤n
∏ i>n
µi × (µn+1 |{R \ ∆n+1 })×
1≤i≤n
∏
×
∏
λi )(An ∩ X) + (
λi )(An ∩ X) = νn (An ∩ X).
i>n+1
Since for an arbitrary n ∈ N the inclusion An ⊂ An+1 holds, we have (∀X)(X ∈ B(RN ) → νn (An ∩ X) ≤ νn+1 (An ∩ X)). Hence there exists a limit lim ν n (X) which we denote by ν∆ (X). n→∞ Establish the following properties of ν∆ . I) The functional ν∆ is countably additive. Let X = ∪k∈N Xk , where (∀m)(∀p)(m ∈ N & p ∈ N & m ̸= p → Xm ∩ Xp = ∅), and, for an arbitrary k ∈ N , let Xk be a Borel subset of the space RN . Then we obtain ∑ ν∆ (∪k∈N Xk ) = ν∆ (Xk ). k∈N
Indeed, on the one hand, we have ν∆ (X) = lim ν n (X) = lim ν n (∪k∈N Xk ) = lim n→∞
n→∞
≤
∑ k∈N
n→∞
lim ν n (Xk ) =
n→∞
∑
∑
ν n (Xk ) ≤
k∈N
ν∆ (Xk ).
k∈N
On the other hand, we have ν∆ (X) = lim
∑
n→∞
ν n (Xk ) = lim ( n→∞
k∈N
+ lim
n→∞
∑
m ∑
k=1
ν n (Xk ) =
k>m
86
ν n (Xk ))+
=
m ∑
ν∆ (Xk ) + lim
n→∞
k=1
i.e.,
∑
∑
ν∆ (∪k∈N Xk ) ≥
ν n (Xk ),
k>m
ν∆ (Xk ).
1≤k≤m
Accordingly, ν∆ (∪k∈N Xk ) ≥
∑
ν∆ (Xk ).
k∈N
i.e., ν∆ (∪k∈N Xk ) =
∑
ν∆ (Xk ).
k∈N
II). The measure ν∆ is nontrivial, since ∏ ν∆ ( ∆i ) = 1. i∈N
III). The measure ν∆ is σ-finite. Indeed, we have RN = (RN \ ∪n∈N An ) ∪ (∪n∈N An ). Since RN \ ∪n∈N An ∈ B(RN ), by the definition of the measure ν∆ we have ν n (RN \ ∪k∈N Ak ) = ν n ((RN \ ∪k∈N Ak ) ∩ An ) = νn (∅) = 0. Since, for an arbitrary natural number n ∈ N , the measure ν n is σ-finite, (n) there exists a countable family (Bk )k∈N of Borel measurable subsets of the space RN such that (n)
a)(∀k)(k ∈ N → ν n (Bk ) < +∞); (n)
b)(∀n)(n ∈ N → An = ∪k∈N Bk ). (n)
Let us consider the family (Bk )k∈N,n∈N . It is clear that (n)
(n)
(∀k)(∀n)(k ∈ N & n ∈ N → ν∆ (Bk ) = ν n (Bk ) < +∞). On the other hand, we have (n)
∪n∈N An = ∪n∈N ∪k∈N Bk , i.e., 87
(n)
RN = (RN \ ∪n∈N An ) ∪ (∪n∈N,k∈N Bk ). The proof is completed. Remark 1. The measure ν∆ described in Lemma 1 can be regarded as an inductive limit of the family (ν)n∈N of invariant measures. Recall that an element h ∈ RN is called an admissible translation (in the sense of invariance ) of the measure ν∆ if (∀X)(X ∈ B(RN ) → ν∆ (X + h) = ν∆ (X)). We define G∆ = {h : h ∈ RN & h is an admissible translation for ν∆ }. It is easy to show that G∆ is a vector subspace of the space RN . Remark 2. The construction of the measure ν∆ belongs to A.B. Kharazishvili (see [69]). Our next theorem gives a representation of the algebraic structure of the vector subspace G∆ of all admissible translations for ν∆ . Theorem 1. The following conditions are equivalent: 1) g = (g1 , g2 , · · ·) ∈ G∆ , 2) (∃ng )(ng ∈ N → the series
∞ ∑
ln(1 −
i=ng
|gi | ) is convergent). bi − ai
Proof: Assume that for an element g = (g1 , g2 , · · ·) ∈ RN the condition 1) is satisfied. Then we have ν∆ (∆ + g) = ν∆ (∆) = 1. On the other hand, we have ν∆ (∆ + g) = ν∆ (∆ + g) = ν∆ (
∏
[ai + gi , bi + gi ]) =
i∈N
= lim ν n (An ∩ (∆ + g)) = lim ( n→∞
×
n→∞
∏ i>n
[ai , bi ]) ∩
∏
∏ 1≤i≤n
µi ×
∏
λi )((
i>n
1≤i≤n
[ai + gi , bi + gi ]) = lim ( n→∞
i∈N
88
∏
∏
1≤i≤n
µi
Ri ×
(
∏
[ai + gi , bi + gi ])) × (
∏
λi ([ai + gi , bi + gi ])) =
i>n
1≤i≤n
= lim
∏
n→∞
λi ([ai , bi ] ∩ [ai + gi , bi + gi ]) = 1.
i>n
Let us show that |gi | = 0). i→∞ |bi − ai |
(∀g)(g = (g1 , g2 , · · ·) ∈ G∆ → lim
Indeed, if we assume the contrary, then there exist a countable subset (nk )k∈N of N and a positive real number ϵ > 0, such that (∀k)(k ∈ N →
|gnk | > ϵ). bnk − ank
Let us choose a number m > 0 such that ϵ · m > 1. Since g ∈ G∆ , we have m · g = (m · g1 , m · g2 , · · ·) ∈ G∆ . In view of the property of σ-additivity of the measure ν∆ , we obtain ν∆ (∆) = ν∆ (∆ + m · g) = 1. But note that (∆ + m · g) ∩ (∪n∈N An ) = ∅. Indeed, assume the contrary and take (xi )i∈N ∈ (∆ + m · g) ∩ (∪n∈N An ). Then it is clear that, for the nk -th coordinate, we have (∃k0 )(k0 ∈ N → (∀k)(k ≥ k0 → (ank + m · gnk ≤ ≤ xnk ≤ bnk + m · gnk ) & (ank ≤ xnk ≤ bnk )). On the other hand, the validity of the condition (∀k)(k ∈ N →
|gnk | > ϵ), bnk − ank
implies the validity of the relation (∀k)(k ∈ N → m · |gnk | > bnk − ank ), 89
which shows us that the inrervals [ank , bnk ] and [ank + gnk , bnk + gnk ] have an i| empty intersection. Hence the condition lim bi|g−a = 0 holds. i i→∞ |gi | i→∞ bi −ai
From the validity of the condition lim
= 0, we conclude that there
exists a natural number ng such that (∀i)(i > ng →
|gi | < 1), bi − ai
since (∀i)(i > ng → λi ([ai , bi ] ∩ [ai + gi , bi + gi ]) =
bi − ai − |gi | |gi | =1− ). bi − ai bi − ai
Keeping in mind that ∏
lim
p→∞
(1 −
i≥ng +p
|gi | )=1 bi − ai
and considering the logarithms of both sides, we have lim
∑
p→∞
This means that the series
ln(1 −
i≥ng +p
∑ i≥ng
ln(1 −
|gi | ) = 0. bi − ai
|gi | bi −ai )
is convergent.
The validity of the implication 1) → 2) is proved. Now let us prove 2) → 1). ∑ i| ln(1 − bi|g−a Let ng be a natural number such that the series ) is converi i≥ng
gent. Let us consider an arbitrary element X having the form ∏ X =B× ∆i , i>n
where B ∈ B(Rn ) (n ∈ N ). The sets of these forms generate the σ-algebra B(An ) of the space An , and the condition B(An ) = B(RN ) ∩ An holds. To prove the implication 2) → 1), it is sufficient to show the validity of the condition ∏ ν∆ (X + g) = ν∆ {[(B × ∆i ) + (g1 , · · · , gng )]× n+1≤i≤ng +n
×
∏
∏
ng +n
[ai + gi , bi + gi ]} = lim
i>ng +n
n→∞
90
i=1
µi (B ×
∏ n+1≤i≤ng +n
∆i )×
×
∏
λi ([ai + gi , bi + gi ] ∩ [ai , bi ]) = ν∆ (B ×
i>ng +n
n→∞
∆i )×
i>n
∏
× lim
∏
(1 −
i>ng +n
∏ |gi | ) = ν∆ (B × ∆i ) = ν∆ (X). bi − ai i>n
We have used the well known result from mathematical analysis (the series
∑
ln(1 −
i≥ng
⇔ lim
n→∞
∏
(1 −
i≥ng +n
|gi | ) is convergent) ⇔ bi − ai
∏ |gi | |gi | )) = ln1 ⇔ lim (1 − ) = 1. n→∞ bi − ai b − ai i i>n +n g
The proof is complete. Remark 3. Let R(N ) be the space of all finite sequences, i.e., R(N ) = {(gi )i∈N |(gi )i∈N ∈ RN &card{i|gi ̸= 0} < ℵ0 }. It is clear that, on∏ the one hand, for an arbitrary compact infinite-dimensional [ak , bk ], we have parallelepiped ∆ = k∈N
R(N ) ⊂ G∆ . On the other hand, G∆ \ R(N ) ̸= ∅, since the element (gi )i∈N defined by (∀i)(i ∈ N → gi = (1 − exp{−
bi − ai } × (bi − ai ))) 2i
belongs to the difference G∆ \ R(N ) . It is easy to show that the vector space G∆ is everywhere dense in RN with respect to the Tikhonov topology, since R(N ) ⊂ G∆ . Definition 1. Let G∆ and G be some vector subspaces of RN such that G∆ ⊂ G ⊂ RN . Let (Gi )i∈I = G/G∆ denote the factor group. We say that a family of elements (gi )i∈I is a selector of a factor group G/G∆ if gi ∈ Gi for each index i ∈ I. We have the following statement. Theorem 2. Assume that G (G∆ ⊂ G ⊂ RN ) is some vector subspace of the space RN .A G-invariant σ-finite Borel measure taking a nonzero value on 91
the element ∆ exists if and only if the cardinality of the factor group G/G∆ is countable. Proof. Let us prove the necessity. Assume that λ is a G-invariant σ-finite Borel measure taking a nonzero value on the element ∆. Assume also that the cardinality of the factor group G/G∆ is uncountable. Let (gξ )ξ 0) → (λ(∆ ∩ (∆ + gξ2∗ − gξ1∗ )) > 0). Applying the scheme proposed in Theorem 3 of Section 11, we can conclude that, for some positive real number q, the following equality is valid: λ|∪n∈N An = q × ν∆ . Accordingly, we have gξ2∗ − gξ1∗ ∈ G∆ , which is a contradiction since the elements gξ1∗ and gξ2∗ belong to different classes of the factor group G/G∆ . Thus (∀ξ1 )(∀ξ2 )(0 < ξ1 < ξ2 < ω1 → λ(gξ1 (∆) ∩ gξ2 ∆) = 0). Now it is easy to construct an ω1 -sequence of disjoint Borel subsets (Kξ )ξ 0. Let us define the functional λ by (∀B)(B ∈ B(RN ) → λ(B) = µ∗ (A ∩ G)). Note that the projection λ1 of the measure λ on the group G is the Ginvariant σ-finite Borel measure defined on the measurable space (G, B(G)) = (G, G ∩ B(RN )). By using one result of H.Weil(see [130]), we conclude that the group G is a locally compact topological vector space. This means that the dimension of the vector space G is finite and we obtain a contradiction to the condition of everywhere density of the group G in RN . In the sequel, we will need some results. 93
Lemma 1. Let N be the set of all natural numbers. Let, for k ∈ N , Sk be the unit circle in the Euclidean space R2 . We may identify the circle Sk with the compact group of all rotations of the Euclidean space R2 . Then, in the system of axioms (ZF )&(DC)&(every subset of R is measurable in the Lebesgue sense), the condition
∏
(∀X)(X ⊂
Sk → X ∈ dom(λ))
k∈N
holds, where λ denotes ∏ the completion of the probability Haar measure λ defined on the compact group Sk . k∈N
The proof of Lemma 1 is considered in Section 7. Lemma 2. For k ∈ N , define the function fk by (∀x)(x ∈ R → fk (x) = (cos(kπx), sin(kπx))). Then in the system of axioms (ZF ) & (DC) the equality ∏ ∏ ∏ fk )(w)), fk )(z) ◦ ( fk )(z + w) = ( (∀z)(∀w)(z ∈ RN & w ∈ RN → ( k∈N
k∈N
holds, where RN denotes the vector space of all real sequences,
k∈N
∏
fk denotes
k∈N
the direct product of the family ∏ of functions (fk )k∈N , ” ◦ ” denotes the usual Sk . group operation in the group k∈N
The proof of the above lemma is analogous to that of Lemma 2 considered in Section 7. Lemma 3. For E ⊂ RN and g ∈
∏
Sk , put
k∈N
{
card((
fE =
∏
fk )−1 (g) ∩ E)
if this is finite,
+∞,
in other cases.
k∈N
Then where h = (
∏
(∀h)(h ∈ RN → fE+h (g) = fE (g ◦ h)), fk )(−h).
k∈N
See the proof of Lemma 3 in Section 7.
94
Lemma 4. In the system of axioms (ZF ) & (DC) & ( every subset of R is measurable in the Lebesgue sense) there exists in RN a translation-invariant measure µ such that: 1) (∀X)(X ⊂ RN → X ∈ dom(µ)); ∏ 2) µ( [− 21i , 21i ]) = 1. i∈N
The proof of an analogous result can be found in Section 7. Remark 5. In the system of axioms (ZF ) & (AC), Lemma 4 is not valid (see e.g. [67]). Now, we can get the validity of the following result. Theorem 5. In the system of axioms
(ZF ) & (DC) & (every subset of R is measurable in the Lebesgue sense) there exists, in the separable Hilbert space ℓ2 , a translation-invariant∏Borel mea[− 21i , 21i ]. sure which takes the value 1 on the infinite-dimensional cube ∆ = i∈N
Indeed, if we put (∀X)(X ∈ B(ℓ2 ) → ν(X) = µ(X)), where B(ℓ2 ) denotes the Borel σ-algebra of subsets of ℓ2 , then we get a proof of Theorem 5. Note that, by using the method considered above, we can construct translationinvariant nontrivial Borel measures in ℓp for an arbitrary natural number p ≥ 1, where ∑ ℓp = {(xk )k∈N : (xk )k∈N ∈ RN → |xk |p < ∞}. k∈N
One can pose the following question: Does there exist a measure µ∆ in RN which would be equivalent to the canonical Gaussian measure defined in RN ? The answer to this question is contained in the following theorem. Theorem 6. For an arbitrary infinite-dimensional parallelepiped ∆ in RN the equality G∆ ̸= ℓ2 holds.
95
Proof.
Let ∆ =
∞ ∏
[ak , bk ] be an arbitrary infinite-dimensional paral-
k=1
lelepiped in RN such that k ∈ N → bk − ak > 0. It is possible to have only two cases: I) The sequence (bk − ak )k∈N is bounded by a positive number m. Then the sequence bk − ak ( )k∈N ∈ ℓ2 , m·k but ∞ ∑ 1 bk − ak |· ) ln(1 − | m·k bk − ak k=n
is not convergent for an arbitrary natural number n. II) The sequence (bk −ak )k∈N be not bounded; then there exists an increasing subsequence (nk )k∈N of natural numbers, such that (∀k)(k ∈ N →
(bnk − ank )2 > 1). k4
Let us define the sequence (xk )k∈N by the formula { b −a nk nk , if m = nk , k2 xm = 0, if m ̸= nk . Then, on the one hand, it is clear that ∑ x2m = +∞, m∈N
which implies (xm )m∈N ∈ / ℓ2 , and, on the other hand, we have ∞ ∑ m=1
ln(1 −
∑ ∑ xnk 1 |xm | ln(1 − ln(1 − 2 ) )= )= bm − am bnk − ank k k≥1
k≥1
which implies (xm )m∈N ∈ G∆ . Remark 5. As shown in Section 6, in some consistent system of axioms it is possible to construct a nontrivial translation-invariant Borel measure µ in RN . We can easily conclude that this measure is not equivalent to the canonical Gaussian measure because the group RN of all admissible translations of µ is wider than the group ℓ2 of all admisible translations of the canonical Gaussian measure defined in RN . 96
The following problem naturally arises: Problem 1. Does there exist a nontrivial translation-invariant Borel measure µ in RN such that the restriction of µ to some Borel subset would be equivalent to the canonical Gaussian measure in RN ? As far as we know, this problem still remains unsolved.
97
7. Invariant Measures in the Nonseparable Banach Space l∞ It is well known that the methods of the Haar measure theory play an important role in various areas of classical and harmonic analysis. In this section we discuss the application of these methods to the theory of invariant measures in the case of the concrete infinite-dimensional nonseparable Banach space. We must say that in some cases the investigation of the concrete property of some effectively defined mathematical object demands applying stronger mathematical methods. Situations may arise where the effectively defined object has different properties in different consistent systems of axioms. We intend to consider such situations below. Let us recall some important notions for Banach spaces, belonging to C.A. Rogers (see [117]). Let U be a Banach space. We say that U has: property P1a: if there is a translation-invariant Borel measure µ on U which is nontrivial in the sense that there is an open set G with 0 < µ(G) < +∞; property P1b: if there is a translation-invariant Borel measure µ on U such that the closed unit ball has measure 1; property P2a: if there is a Borel measure µ on U which is nontrivial in the sense that there is an open set G with 0 < µ(G) < +∞ and for which the measure of a ball depends only on the ball radius; property P2b: if there is a Borel measure µ on U for which the closed unit ball has measure 1 and the measure of any ball depends only on the ball radius. In [117] Rogers posed the question which Banach spaces have properties P1a–P2b. Some interesting results were obtained (see [22], [82]) in this context. We will consider Rogers’ problem for the nonseparable Banach space l∞ of all bounded real sequences (xk )k∈N with the standard norm || · ||∞ defined by ||(xk )k∈N ||∞ = sup |xk |. k∈N
In connection with the above-mentioned properties, the main result obtained for l∞ is formulated as: Theorem 1. (C.A. Rogers) In the system of axioms (ZF )&(AC)&(c is of measure zero), the nonseparable Banach space l∞ does not have properties P1a, P1b, P2a and P2b. 98
Let us consider Rogers’ problem in the system of axioms: (ZF ) & (DC) & (every subset of R is measurable in the Lebesgue sense). The choice of such a system of axioms for investigating the properties of l∞ is justified by the following Theorem 2 (R.M. Solovay). If the system of axioms: (ZF ) & (AC) & (there exists an uncountable inaccessible cardinal number) is consistent, then the system of axioms: (ZF ) & (DC) & (every subset of R has a Baire property and is measurable in the Lebesgue sense) is also consistent. The proof of Theorem 2 can be found in [63]. In connection with Theorem 2, we must say that the following theorem is interesting. Theorem 3 (Michelsky-Sverkovsky). In the system of axioms (ZF ) & (AD), every subset of R is measurable in the Lebesgue sense. The proof of Theorem 3 is given in the Appendix. Let us formulate the main rezult of this section. Theorem 4. In the system of axioms: (ZF ) & (DC) & (every subset of R is measurable in the Lebesgue sense), the Banach space l∞ has property P 1b. The proof of Theorem 4 is based on the following lemmas. Lemma 1. Let N be the set of all natural numbers. Let, for k ∈ N , Sk be the unit circle in the Euclidean space R2 . We may identify the circle Sk with a compact group of all rotations of the Euclidean space R2 . Then in the system of axioms (ZF ) & (DC) & (every subset of R is measurable in the Lebesgue sense) 99
the condition (∀X)(X ⊂
∏
˜ Sk → X ∈ dom(λ))
k∈N
˜ denotes the completion of the probability Haar measure λ defined holds, where λ ∏ on the compact group Sk . k∈N
Proof. Let N be the set of all natural numbers . For every index i ∈ N , we put Ei = {0; 1}, Di = P (Ei ). Next we define the measure µi on Di by the formula card(X) µi (X) = . 2 Let us denote by µ the measure µi (i ∈ N ). It is clear that ((Ei , Di , µi ))i∈N is a family of probability spaces. By the symbol ∏ ∏ µi ) Di , ({0; 1}N , i∈N
i∈N
we denote the product probability space ∏ of this family. µi . Let us denote by µN the measure i∈N
Let f be the function from {0; 1}N into [0; 1] defined by the formula f (x) =
∑ xn 2n+1
((xn )n∈N ∈ {0; 1}N ).
n∈N
This function is not a bijection but it is a surjection and its restriction to the set D = {x ∈ {0; 1}N : (∀n)(∃m > n) (xm = 1)} is a bijection onto ]0; 1]. The complement of D to {0; 1}N is countable; hence there exists a bijection g : {0; 1}N → [0; 1[ such that card({x ∈ {0; 1}N : f (x) ̸= g(x)}) = ℵ0 . This bijection gives us an isomorphism between the product measure space (({0; 1}N , B({0; 1}N ), µN ) and the probability space ([0; 1[, B([0; 1[), b1 ), where b1 is the restriction of the Lebesgue measure to the class of all Borel subsets of the interval [0; 1[. 100
Denote by ≡ relation of a the Borel isomorphism between two measures. It is clear that b1 ≡ µN . On the one hand, we have N bN 1 ≡µ .
On the other hand, we have b1 ≡ µN ≡ µN ×N ≡ bN 1 ≡ λ. This means that the standard Borel measure b1 is Borel isomorphic to the probability Haar measure λ. Denote this isomorphism by φ. Let us consider an arbitrary set W ⊂ SkN . It is clear that φ−1 (W ) = X ∪ Y, where 1) X ∈ dom(b1 ), 2) (∃Z)(Y ⊂ Z ∈ B([0; 1]) & b1 (Z) = 0). Let us consider the set X ∪ Z. From the isomorphism between the measures b1 and λ we have λ(φ(Z \ X)) = 0. On the one hand, we may write W = φ(X) ∪ φ(Y ). On the other hand, we have φ(Y ) ⊂ φ(Z). Clearly, λ(φ(Y )) = 0 since λ(φ(Z)) = 0. This completes the proof of Lemma 1. Lemma 2. For k ∈ N , define the function fk by (∀x)(x ∈ R → fk (x) = (cos(πx), sin(πx)). Then in the system of axioms (ZF ) & (DC) the equality ∏ ∏ ∏ (∀z)(∀w)(z ∈ RN & w ∈ RN → ( fk )(z + w) = ( fk )(z) ◦ ( fk )(w)) k∈N
101
k∈N
k∈N
∏
holds, where RN denotes the vector space of all real sequences,
fk denotes
k∈N
the direct product of the∏family of functions (fk )k∈N , ” ◦ ” denotes the group operation in the group Sk . k∈N
Proof. Note that, for an arbitrary y ∈ RN , we have a unique representation such that (0) (1) y = (yk )k∈N + (yk )k∈N , (1)
(0)
where (yk )k∈N ∈ (2Z)N and 0 < yk < 2 (k ∈ N ). Let us consider such a representation of the elements Z, W, Z + W . This is (0)
(1)
Z = (Zk )k∈N + (Zk )k∈N . (0)
(1)
W = (Wk )k∈N + (Wk )k∈N . (0)
(1)
Z + W = ((Z + W )k )k∈N + ((Z + W )k )k∈N . It is clear that (1)
(1)
(1)
(1)
(∀k)(k ∈ N → fk (Zk + Wk ) = fk (Zk ) ◦k fk (Wk ), where ◦k denotes the group operation in the group Sk . The number 2 is the smallest period of the function fk . Finally, we have (0)
(0)
fk (Zk + Wk
(1)
(1)
(1)
(1)
(1)
This means that ∏ ∏ ∏ fk )(W ). fk )(Z) ◦ ( fk )(Z + W ) = ( ( k∈N
k∈N
k∈N
Lemma 2 is proved. Lemma 3. For E ⊂ RN and g ∈
∏
Sk , put
k∈N
{
card((
fE =
∏
fk )−1 (g) ∩ E), if this is finite;
k∈N
+∞,
in other cases.
Then where h = (
∏
(1)
+ Zk + Wk ) = fk (Zk + Wk ) = fk (Zk ) ◦k fk (Wk ).
(∀h)(h ∈ RN → fE+h (g) = fE (g ◦ h)), fk )(−h).
k∈N
Proof. By Lemma 2,we conclude that 102
card((
∏
k∈N ∏
fk )−1 (g) ∩ (E + h)) =
card((( fk )−1 (g) − h) ∩ E) = k∈N ∏ ∏ card(( fk )−1 (g ◦ ( fk )(−h)) ∩ E). k∈N
k∈N
This completes the proof of Lemma 3. Lemma 4. In the system of axioms: (ZF ) & (DC) & ( every subset of R is measurable in the Lebesgue sense) there exists in RN a translation-invariant measure µ such that: 1) (∀X)(X ⊂ RN → X ∈ dom(µ)); 2) µ([−1; 1]N ) = 1. Proof. Let us define the functional µ by ∫ (∀E)(E ⊂ RN → µ(E) =
∏
fE (g)dλ(g)). Sk
k∈N
∑ The functional µ is a measure since f∅ (g) = 0 and f∪k∈N Ek (g) = k∈N fEk (g),where (Ek )k∈N is an arbitrary family of disjoint subsets of RN . The functional µ is a translation-invariant measure. Indeed, for an arbitrary h ∈ RN , from the invariance of the measure λ and by Lemma 3, we have ∫ µ(E + h) = fE+h (g)dλ(g) = ∏ Sk
k∈N
∫ =
∏
∫
fE (g ◦ h)d(λ(g ◦ h)) = Sk
k∈N
∏
fE (g ◦ h)dλ(g ◦ h) = Sk ◦h
k∈N
∫ =
∏
fE (s)dλ(s) = µ(E). Sk
k∈N
Note that (∀g)(g ∈
∏
Sk → f[−1;1[N (g) = 1).
k∈N
This implies that µ([−1; 1[N ) = 1. It is also clear that [−1; 1]N \ [−1; 1[N = ∪k∈N Xk , 103
where Xk = {1}k × [−1; 1]N \{k} and µ(Xk ) = 0. Lemma 4 is proved Remark 1. In the system of axioms (ZF ) & (AC), Lemma 4 is not valid (see [47], p.65) Now the proof of Theorem 4 is obvious. Indeed, if we put (∀X)(X ∈ B(l∞ ) → ν(X) = µ(X)), where B(l∞ ) denotes the Borel σ-algebra of subsets of l∞ , then we obtain the proof of Theorem 4. Following Fremlin, a measure µ defined on a topological space is effectivelly locally finite if, whenever µ(E) > 0, there is a measurable open set G of finite measure, such that µ(E ∩ G) > 0. The next theorem is the object of interest in the context of Theorem 4. Theorem 5 (D. Fremlin). In the system of axioms: (ZF ) & (AC) & (c is not of measure zero), l∞ has the properties P 2a and P 2b. The proof of Theorem 5 can be found in [34], p.6. Let us briefly consider the construction of the measure µ, belonging to D.Fremlin. Let λ be the usual Radon probability measure on [0; 1]N . Let k be an atomlessly-measurable cardinal and ν a normal measure on k. Then there is a function ψ : k → [0; 1]N . Write ν for νψ −1 , so that ν is an extension of λ defined on every subset of [0; 1]N . For any Borel set E ⊆ l∞ , define fE : [0; 1]N → [0; ∞] by setting { card{z : z ∈ Z N , 2(x + z) ∈ E}), if this is finite; fE = +∞, in other cases. Note that the measure µ defined by (∀E)(E ∈ B(l∞ ) → µ(E) =
∫ fE (x)ν(dx)) [0;1]N
is effectivelly locally finite, and the proof of Theorem 5 is complete. We say that the measure µ defined on the Banach space has property P 2b∗ if the measure of an arbitrary union of disjoint balls with equal radiuses is invariant under the translations. Note that property P 2b∗ is stronger than the property 104
that the measure of a ball depends only on the ball radius,and is weaker than the property of translation-invariance. Theorem 6. The above measure µ has no property P 2b∗ and, correspondingly, is not a translation-invariant measure. Proof. Assume the contrary, and let the measure µ have the property P 2b∗ . Let φ be a bijection between [−1; 1]N and {0; 12}N . Let us consider the set Y = (φ−1 (z) + z)z∈{0;12}N . It is clear that Y ∈ dom µ and (∀Y0 )(Y0 ⊂ Y → Y0 ∈ dom µ). If we suppose that 1 1 B( ) = {x : ||x|| < }, k k then
1 Y0 = ∩k∈N (∪z∈{z:φ−1 (z)+z∈Y0 } {φ−1 (z) + z + B( )}). k Let G be an arbitrary open set with 0 < µ(G) < +∞ and µ(Y ∩ G) > 0. Then we obtain Y ∩ G = ∪k∈N Yk ,
where Yk = {z+φ−1 (z) : z ∈ {0; 12}N & min{m : m ∈ N &z + φ(z)−1 + B( It is clear that µ(Y ∩ G) =
∑
µ(Yk ).
k∈N
From the condition µ(Y ∩ G) > 0 we have (∃k0 )(k0 ∈ N → µ(Yk0 ) > 0). Let us consider the open set Yk0 + B(
1 ) ⊂ G. k0
We have 105
1 ) ⊂ G} = k}. m
Yk0 = {z + φ−1 (z)}z∈Y˜k , 0
where Y˜k0 ⊂ {0; 12} . 1 Let us consider Yk0 + B( 12·k ) being an union of disjoint balls with equal 0 radiuses. It is clear that N
µ(Yk0 + B(
1 )) > 0. 12 · k0
Consider the set ∪g∈{0; 4k1
0
}N (Yk0
+ B(
1 ) + g). 12 · k0
As the measure µ has property P 2b∗ and (Y0 + B(
1 ) + g)g∈{0; 4k1 }N 0 12 · k0
1 is the family of disjoint translations of Yk0 + B( 12·k ) we have 0
µ(∪g∈{0; 4k1
0
}N (Y0
+ B(
1 ) + g)) = +∞, 12 · k0
but this is impossible since the set constructed above is a subset of the set G. This means that the measure µ is not effectively locally finite and we get a contradiction. This completes the proof of Theorem 6. Remark 2. The measure in Theorem 4 is a translation-invariant measure and, therefore, has the property P 2b∗ . Using the same arguments as in Theorem 6, it is not difficult for one to prove that this measure is not effectively locally finite .
106
8. Vector Fields of velocities in the Infinite-Dimensional Topological Vector Space RN which preserve the Measure µ In this section we want to remind the reader some notions and elementary facts from measure theory and the theory of differential equations. Let I be a nonemty countable set of parameters. Definition 1. The family {g t }t∈R of transformations of RI is called the one-parameter group if (∀s)(∀t)(s ∈ R & t ∈ R → g t+s = g t · g s ) and g 0 is the identity transformation of the space RI . Definition 2. (RI , {g t }t∈R ) is called the phase flow if {g t }t∈R is a oneparameter group of transformations of the space RI . Definition 3. A transformation g : R × RI → RI is called a one-parameter group {g t }t∈R of diffeomorphisms of RI if the following four conditions are fulfilled: 1) g(t, x) = g t x for all t ∈ R and all x ∈ RI ; 2) g is a differentiable transformation on R × RI ; 3) g t : RI → RI is a diffeomorphism for t ∈ R; 4) the family {g t }t∈R is an one-parameter group of transformations of RI . Let (RI , {g t }t∈R ) be the phase flow defined by the one-parameter group of diffeomorphisms of RI . Definition 4. The velocity of a phase point defined by v(x) =
d t g (x)|t=0 dt
is called the phase velocity v(x) of the phase flow g t at the point x ∈ RI . Definition 5. Let µ be a measure defined on some {g t }t∈R -invariant σalgebra of subsets of the space RI . We say that the phase flow (RI , {g t }t∈R ) preserves the measure µ if (∀t)(∀B)(t ∈ R & B ∈ dom(µ) → µ(g t (B)) = µ(B)). Theorem 1. Let A = (aij )i,j∈1,n |det(A)| = 1. Then
be
an n-dimensional matrix with
(∀B)(B ∈ B(Rn ) → ℓn (A(B)) = ℓn (B)), 107
where B(Rn ) denotes the Borel σ-algebra of subsets of Rn and ℓn denotes the n-dimensional Lebesgue measure defined in Rn . Proof. By the well-known formula of substituting variables in multiple integrals, we have ∫ (∀B)(B ∈ Rn →
∫ dx1 × · · · × dxn = B
A(B)
where
|
D(x1 , · · · , xn ) |dξ1 × · · · × dξ2 ), D(ξ1 , · · · , dξn )
x1 = x1 (ξ1 , · · · , ξ2 ), .. . xn = xn (ξ1 , · · · , ξn ),
and D(x1 , · · · , xn ) ∂xi = det(( )1≤i≤n,1≤j≤n ). D(ξ1 , · · · , ξn ) ∂ξj From the equality |
D(x1 , · · · , xn ) | = |det(A)| D(ξ1 , · · · , ξn )
it immediately follows that (∀B)(B ∈ B(Rn ) → ℓn (A(B)) = ℓn (B)). Let µ = λ∆ , where λ∆ is the measure constructed in Section 6 for ∆ = [0; 1[N . The following result is valid. Theorem 2. Let Dn be the group of all n-dimensional A = (aij )i,j∈1,n matrices with |det(A)| = 1. Let us denote G = ∪n∈N (Dn × I (n) ), where I (n) denotes the identity transformation of the topological vector space RN \{1,···,n} . Then the measure µ is G-invariant. Proof. Note that,for an arbitrary natural number n, the measure µ can be considered as the product of two measures µ1 and µ2 , where µ1 coincides with the n-dimensional Lebesgue measure. The measure µ1 is Dn -invariant (see Theorem 1). The measure µ2 can be considered as an I (n) -invariant measure. By 108
using the well-known result on the product of σ-finite invariant Borel measures, we conclude that the measure µ = µ1 × µ2 is Dn × I (n) -invariant and Theorem 2 is proved. Theorem 3. Let A = (aij )i,j∈1,n be an n-dimensional matrix. Then det(et·A ) = et·T r(A) , where T r(A) =
n ∑
aii .
i=1
The proof of Theorem 3 can be found e.g.in [1]. Definition 6. We say that there exists a vector field A defined on RI if the value of the vector quantity A is specified at each point x of RI , i.e. A = A(x). It is clear that an arbitrary family of transformations {g t }t∈R of RI can be considered as a vector field defined on the corresponding vector space. If {g t }t∈R is the family of differentiable transformations, then (RI , {g t }t∈R ) is called a vector flow on RI . We will deal with a stationary field which does not change as time passes. If such a variation takes place, we will consider the field at a fixed moment of time and thus reduce our consideration to a stationary field. As examples of vector fields, we can consider a field of velocities v, a field of momentum density ρv (where ρ is the mass distribution density) for a liquid or gas flow, a field of force F , an electric field E (where E is electric field strength), etc. Definition 7. The vector field defined by (∀x)(x ∈ RI → v(x) =
d t g (x)|t=0 ). dt
is called the vector field of velocities v on RI determined by the phase flow (RI , {g t }t∈R ). Definition 8. We say that a vector field of velocities v defined on RI preserves the measure µ with dom(µ) = B(RI ) if (∀B)(∀t)(B ∈ dom(µ) & t ∈ R → g t (B) ∈ dom(µ) & µ(g t (B)) = µ(B)). Definition 9. The divergence of the vector field of velocities v on RI is denoted by div(v) and defined by div(v) =
∑ ∂vi , ∂xi i∈I
109
where (vi )i∈I is the family of components of the vector velocity v and xi ∈ R for all i ∈ I. Theorem 4.( Liouville) Let v = A(x) be a linear continuously differentiable vector field of velocities defined on Rn , and D(0) be some Borel subset of Rn . Then the formula ∫ dℓn (Dt ) = div(A)dℓn dt D(t)
is valid, where D(t) denotes the state of the subset D(0) at the moment t under the action of the phase flow defined by the vector field of velocities v = A(x). The proof of an equivalent form of Theorem 4 is given, for instance, in the Appendix. Example 1. The force acting between the constituents (electrons and nuclei) of matter is given by Coulomb’s inverse square law of electrostatics: If two particles have charges q1 and q2 and locations x1 and x2 in R3 , then F1 -the force on the first due to the second–minus F2 - the force on the second due to the first, and is given by −F2 = F1 = q1 q2
(x1 − x2 ) . |x1 − x2 |3
If q1 q2 < 0, then the force is attractive; otherwise, it is repulsive. This force can be written as minus the gradient (denoted by ∇) of a potential energy function 1 W (x1 , x2 ) = q1 q2 , |x1 − x2 | i.e., F1 = −∇1 W and F2 = −∇2 W. If there are N electrons located at X = (x1 , · · · , xN ) with xi ∈ R3 , and k nuclei with positive charges Z = (z1 , · · · , zk ) and located at R = (R1 , · · · , Rk ) with Ri ∈ R3 , then the total-potential energy function is W (X) = −A(X) + B(X) + U with A(X) = e2
N ∑
V (xi ),
i=1
V (x) =
k ∑
zj |x − Rj |−1 ,
j=1
110
∑
B(X) = e2
|xi − xj |−1 ,
1≤im
∑
tai
= lim e1≤i≤m m→∞
∑ = ei≥1
×(
∏
∏
[0; etai [) =
i>m
µi )(B) × lim ( m→∞
1≤i≤n tai
×(
∏
[0; 1[i ))×
n+1≤i≤m
1≤i≤m
λi (
∏
∏
i≥m+1
µi )(B) = µ(B ×
1≤i≤n
λi )( ∏
∏
[0; etai [) =
i≥n+1
[0; 1[k ),
k>n
where Am is the m-dimensional matrix standing in the upper-left part of the matrix A. The sufficiency is proved. 116
Remark 4. If A is an infinite-dimensional diagonal matrix and T r(A) is absolutely convergent, then, using the method considered above, we obtain (∀B)(B ∈ B(RN ) → µ(etA (B)) = etT r(A) × µ(B)). Thus, we get the validity of the following result which is a direct analog of Liouville’s theorem (see Theorem 4.) Theorem 10. Let v = Ax be a vector field of velocities defined on RN , where A is an infinite-dimensional diagonal matrix such that T r(A) is absolutely convergent. Let D(0) be some Borel subset in RN . Then the formula ∫ dµ(D(t)) = div(A)dµ dt D(t)
is valid, where D(t) denotes the state of the subset D(0) at the moment t under the action of the phase flow (RN , etA × (·)). Example 2. Let (fk )k∈N be a family of continuous functions defined in R such that t ∑ ∫ (∀t)(t ∈ R → | fk (τ )dτ | < ∞). k∈N
0
Let us consider the system of differential equations { dx2k+1 = fk (t)x2k−1 , dt dx2k dt = −fk (t)x2k (k ∈ N ),
(6)
for k ∈ N . Then the vector flow defined by the vector field of velocities (6) preserves the measure µ. Example 3. Let (fk )k∈N be a family of continuously differentiable functions defined in R2 , such that ∑ k∈N
sup |
∂ 2 fk | < +∞. ∂x∂y
Let us consider the system of differential equations { ∂x ∂fk 2k−1 = ∂x (x2k−1 , x2k ), ∂t 2k ∂fk ∂x2k ∂t = − ∂x2k−1 (x2k−1 , x2k ), for k ∈ N . 117
(7)
Then the vector flow defined by the vector field of velocities (7) preserves the measure µ. Theorem 11. Let an operator U : RN → RN be defined by U (x1 , x2 , · · ·) = (c1 x1 , c2 x2 , · · ·), where (∀k)(k ∈ N → ck > 0) and
∏
ck = 1.
k∈N
Then the measure µ is invariant under the operator U if and only if
∏
min{ck , 1}
k∈N
is convergent. Proof. Since ck = eln(ck ) , all conditions of Theorem 7 hold. It is easy to verify that in our ∏ asumptions the following two conditions are equivalent: 1) The series min{ck , 1} is convergent to zero. k∈N ∑ ln(ck ) is absolutely convergent. 2) The series k∈N
Theorem 12. Let us consider the differential equation dΨ = AΨ + b, dt {
where A = (aij )i,j∈N , ai,j = ∑
(8)
di , if i = j, 0, if i ̸= j,
di = 0, di ̸= 0 (i ∈ N ), t ∈ R, Ψ ∈ RN , b = (b1 , b2 , · · ·) ∈ RN .
i∈N
Then: At 1) the measure µ is invariant under the family ∑ of transformations (e × −1 dk is absolutely convergent (·) − A × b)t∈R defined by (8) if the series k∈N ∑ and there exists a natural number n0 such that the series ln(1 − |d−1 k bk |) is k≥n0
convergent; 2) if the measure µ is invariant under the family of transformations described above, then the following two conditions are equivalent: ∑ I ) the series dk is absolutely convergent, k∈N ∑ II ) there exists a natural number n0 such that the series ln(1−|d−1 k bk |) k≥n0
is convergent. Proof. Using Theorem 7 of this section and Theorem 1 of Section 6, we can easily prove part 1). 118
Proof of part 2). Let the measure µ be invariant under the family of transformations mentioned above. Let condition I ) be satisfied. Let us consider an arbitrary Borel subset B ∈ B(RN ). Then, for e−At (B), we obtain µ(etA (e−tA (B)) − A−1 × b) = µ(B − A−1 × b) = µ(e−tA (B)). By using Theorem 7, we conclude that µ(B + A−1 × b) = µ(B) for B ∈ B(RN ). It means that the translation A−1 × b = (d−1 i bi )i∈N is an admissible translation for the measure µ. Using Theorem 1 from Section 6, we conclude that there exists n0 ∈ N such that the series ∑ ln(1 − |d−1 i bi |) i≥n0
is convergent. Now, let condition II) be satisfied. Then, for an arbitrary B ∈ B(RN ), we obtain µ(etA (B) − A−1 b) = µ(etA (B)) = µ(B). ∑ di is absolutely convergent. By Theorem 7, we conclude that the series i≥1
Example 4. Let d1 = b1 = 1 and (∀k)(k ≥ 2 → −dk = 2k bk = 2−k+1 ). Then all conditions of part 1 of Theorem 12 are satisfied and the measure µ is invariant under the family of transformations defined by (8).
119
9. Invariant Borel Probability Measures on the Unit Sphere in an Infinite-Dimensional Separable Complex-Valued Hilbert Space Let us introduce some notation: C = { z = x + iy : x ∈ R & y ∈ R} is a complex number space. x is called the real component of a complex number z = x + iy and denoted by Re(z). y is called an imaginary component of a complex number z = x + iy and denoted by Im(z). CN = k ∈ N.
∏
Ck is the product of complex number spaces, where Ck = C for
k∈N
∑
W 2 = {(zk )k∈N : zk ∈ C, k ∈ N,
|zk |2 < +∞} is an infinite-dimensional
k∈N
separable complex-valued Hilbert space equipped with the usual inner scalar product (·, ·), where (∀(zk )k∈N )(∀(wk )k∈N )((zk )k∈N ∈ W 2 & (wk )k∈N ∈ W 2 → ∑ zk × wk ), → ((zk )k∈N , (wk )k∈N ) = k∈N
where wk denotes the conjugate of the number wk (k ∈ N ). S 2 = {(zk )k∈N : (zk )k∈N ∈ W2 &
∑
|zk |2 = 1} is the unite sphere in W 2 .
k∈N
(ek )k∈N is a complete family of orthonormal vectors in W 2 . B(W 2 ) is the σ-algebra of all Borel subsets in W 2 . B(S 2 ) is the σ-algebra of all Borel subsets in S 2 . Rn is an n-dimensional Euclidean vector space. B(Rn ) is the σ-algebra of all Borel subsets of Rn . Let us consider the following well-known simple result. Theorem 1. On the unit sphere S 2 , there does not exist a probability Borel measure invariant under the group of all unitary operators of the space. 120
Proof. Assume the contrary and let p be such a measure. Denote by (Ψk )k∈N√the family of full orthonormal vectors in W 2 . It is clear that ||Ψk − Ψn || √= 2. Let U (Ψ, r) be a spherical neighbourhood of a point Ψ ∈ S 2 with r < 22 . Note that p(U (Ψ, r)) = 0 for arbitrary Ψ ∈ S 2 . Indeed, if we assume the contrary, then from the invariance under the group of all unitary operators of the measure p, we conclude that (U (Ψk , r))k∈N is a family of disjoint Borel subsets with an equal positive p-measure in S 2 , such that ∑ 1 = p(S 2 ) ≥ p(∪k∈N U (Ψk , r)) = p(U (Ψk , r)) = +∞. k∈N
Let M be a countable everywhere dense subset in S 2 . It is clear that S 2 = ∪Ψ∈M U (Ψ, r). On the one hand, we have p(S 2 ) = 1, which is not true because ∑ p(S 2 ) = p(∪Ψ∈M U (Ψ, r)) ≤ p(Ψ, r) = 0. Ψ∈M
We obtain the contrary and Theorem 1 is proved. Let, for k ∈ N , rk denote a positive number such that ∫ ∫ 1 1 − x21 +x22 2 dx1 dx2 = e− 2k . e 2 2π x21 +x22 ≤rk Let, for k ∈ N , µk denote a two-dimensional Gaussian measure in Rk2 whose density has the form x2 +x2 1 − 12δ2 2 k , e 2πδk2 where Rk = R, δk =
1 . 2k r k
In the sequel, we need the following results. Lemma 1. σ(Cl(CN ) ∩ (∪k∈N An )) = σ(Cl(W 2 ) ∩ (∪n∈N An )) = = B(CN ) ∩ (∪n∈N An ) = B(W 2 ∩ (∪n∈N An )), where Cl(·) and B(·) denote respectively cylindrical and a Borel σ-algebra of subsets of the corresponding spaces. Lemma 2. Let E be a base space, S be a G-invariant algebra of subsets of E, and µ be a G-invariant probability measure defined on S. Then the extension µ of the measure µ to the σ-algebra σ(S) is a G-invariant probability measure. 121
Lemma 3. Let (Ei , Si , Gi , µi )1≤i≤n be a family of invariant measurable ∏ ∏ spaces with invariant σ-finite measures. Then µi is a Gi -invariant 1≤i≤n
measure.
1≤i≤n
Theorem 2. Let (∀X)(X ∈ B(Rk2 ) → µck (X c ) = µk (X)), where X c = {x + iy : (x, y) ∏ ∈ X}. Then, for the measure µck , the following conditions hold: k∈N ∏ c 1) µk (W 2 ) = 1; k∈N ∏ c 2) The measure µk is invariant under the transformation A : W 2 → k∈N
W 2 whose corresponding matrix in the standard basis is a diagonal matrix (akm )k,m∈N with akk = eidk , k ∈ N, dk ∈ R. Proof. Let
B0,2−k = {(x, y) : (x, y) ∈ Rk2 & x2 + y 2 ≤ 2−2k }, (∀n)(n ∈ N → An =
n ∏
Ck ×
k=1
∏
B0,2−k ).
k>n
It is clear that (∀n)(n ∈ N → An ∈ B(W 2 )). Hence ∪n∈N An ⊂∏ W 2. µcn (∪n∈N An ) = 1. Let us show that n∈N
It is clear that (∀n)(n ∈ N → An ⊂ An+1 ). By the property of lower semicontinuity of probability measures, it is sufficient to show that limn→∞ µ(An ) = 1. Indeed, using the property of continuity from the left of the function ex at the point x = 0, for an arbitrary positive number ϵ we conclude that there exists δ > 0 such that (∀x)(−δ ≤ x ≤ 0 → 1 − ϵ ≤ ex ≤ 1). Let nϵ be a natural number such that −δ ≤ −
∑ 1 ≤ 0. 2k
k≥nϵ
Then, for n ≥ nϵ , we obtain 122
∏
µck (An ) =
k∈N
n ∏
µck (
k=1
=
n ∏
Ck ) ×
−
e
µck (
k>n
k=1
∏
∏
−
1 2k
=e
∑
∏
B0,2−k ) =
k>n 1 2k
.
k>n
k>n
Hence
∏
1−ϵ
n
the transformation A for all X ∈ B(Cn ). Indeed, ∏ ∏ ∏ ∏ µck (A(X × B0,2−k )) = µck (An (X) × eidk (B0,2−k )) = k∈N
=
∏ k∈N
k>n
µck (An (X) ×
∏ k>n
k∈N
B0,2−k ) =
n ∏
k>n
µck (An (X)) ×
k=1
123
∏ k>n
µck (
∏
k>n
B0,2−k ) =
=
n ∏
µck (X) ×
k=1
∏
µck (
k>n
∏
B0,2−k ) =
k>n
∏
∏
µck (X ×
k∈N
B0,2−k ),
k∈N
where An = (akm )k,m∈1,n is an n-dimensional diagonal matrix with akk = eidk , 1 ≤ k ≤ n. Theorem 2 is proved. Let X ∈ B(S 2 ). We say that a set IX ⊆ W 2 is the X-cone if IX = {Ψ : Ψ ∈ W 2 & (∃α)(α ∈ R+ → αΨ ∈ X)}. Lemma 4. (∀X)(X ∈ B(S 2 ) → IX ∈ B(W 2 )) and the class K(W 2 ) of all conical subsets in W 2 is an A-invariant σ-algebra of subsets of W 2 \ {0}. Proof. Let us denote by G the class of all open sets in S 2 . It is clear that σ(G) = B(S 2 ). Let us show that, for arbitrary Y ∈ G, the Y -cone IY ∈ B(W 2 ). Indeed, let z0 ∈ IY . This means that there exists α ∈ R+ such that αz0 ∈ Y. There are two possible cases: α ≤ 1 or α > 1. Case 1. If α ≤ 1, then |z0 | ≥ 1 and there exists r (0 < r < 21 ) such that {z : z ∈ S 2 & |z − z0 | ≤ r} ⊂ Y. Now {z : z ∈ W 2 & |z − z0 | < r} ⊂ IY . Case 2. If α > 1, then |z0 | < 1 and {z : z ∈ W 2 & |z − z0 |
)k∈N ∈ X}) = µ(X)), where < ·, · > is the usual inner product in L2 (R3 , C) defined by ∫ (∀u1 )(∀u2 )(u1 , u2 ∈ L2 (R3 , C) →< u1 , u2 >= u1 × u2 dx1 dx2 dx3 ). R3
It is clear that the probability measure ν is defined on the Borel σ-algebra of subsets of the unit sphere S ∗2 in L2 (R3 , C). The matrix defined by operator H coincides with the infinite diagonal matrix (aij )i,j∈N in the basis (Ψk )k∈N i so that aii = λi . Hence e− h¯ tH = (bmn )m,n∈N is also a diagonal matrix with i bmm = e− h¯ λm (m ∈ N ). i Let us show that the measure ν is invariant under the phase flow (S ∗2 , (e− h¯ tH )t∈R ) defined by (1). Indeed, for arbitrary X ∈ B(S 2 ), we have ν(e− h¯ tH {Ψ : (< Ψ, Ψk >)k∈N ∈ X}) = ν({e− h¯ tH Ψ : (< Ψ, Ψk >)k∈N ∈ X}) = i
i
= ν({Ψ∗ : (< e h¯ tH Ψ∗ , Ψk >)k∈N ∈ X}) = ν({Ψ∗ : (< Ψ∗ , Ψk >)k∈N ∈ e− h¯ tH (X)}) = i
i
= µ(e− h¯ tH (X)) = µ(X) = ν({Ψ : (< Ψ, Ψk >)k∈N ∈ X}). i
Theorem 6 is proved. Corollary 2. By the Poincar´e-Carath´eodory theorem, we conclude that ν-almost every point Ψ ∈ S ∗2 is stable in the sense of Poisson; moreover, (∀Y )(Y ∈ B(S ∗2 ) → ν({Ψ : Ψ ∈ Y & (∀t)(t > 0 → e− h¯ tH Ψ ∈ / Y )}) = 0). i
Corollary 3. By the Birkhoff-Chintchin theorem, we conclude that, for an arbitrary ν-integrable function f (Ψ) defined in S ∗2 , there exists a limit n−1 i 1∑ f (e− h¯ tH ψ) = f ∗ (Ψ) (mod ν) n→∞ n
lim
k=0
126
ν-almost everywhere in S ∗2 such that f ∗ (e− h¯ tH Ψ) = f ∗ (Ψ) (mod ν) i
and
∫
∫
∗
f (Ψ)dν(Ψ) = S ∗2
f (Ψ)dν(Ψ). S ∗2
The reader can acquire other interesting information on the Schr˝odinger equation in many works and monographs; see e.g.[88].
127
10. Invariant Borel Measures in an Infinite-Dimensional Separable Complex-Valued Hilbert Space Let us introduce some notions 1 }&Ci = C). 2i ∏ ∏ (∀n)(n ∈ N → An = Ci × ∆i ).
(∀i)(i ∈ N → ∆i = {z : z ∈ C&|z|
n
Let µi denote a two-dimensional classical Borel measure defined in C and taking value 1 on the set ∆i . Analogously, let λi be a two-dimensional normed classical Borel measure defined on ∆i . For arbitrary n ∈ N , let us denote by νn the measure defined by ∏ ∏ νn = λi , µi × 1≤i≤n
i>n
and by νn the Borel measure in W 2 defined by (∀X)(X ∈ B(W 2 ) → νn = νn (X ∩ An )). By the scheme considered in Section 6 we can obtain the validity of the following results. Lemma 1. For an arbitrary Borel set X ⊆ W 2 , there exists a limit ν∆ (X) = lim νn (X). n→∞
Moreover, the functional ν∆ is a nontrivial σ-finite measure defined on the Borel σ-algebra B(W 2 ). We say that the measure ν∆ is invariant under a transformation U : W 2 → W if 2
(∀X)(X ∈ B(W 2 ) → ν∆ (U (X)) = ν∆ (X)). Let us put G0 = {eiA : A = (akm )k,m∈N is a diagonal matrix with positive diagonal elements}. Let us denote by G the group of transformations of W 2 dgenerated by G0 . The following proposition is valid.
128
Theorem 1. The measure µ∆ is a G-invariant measure taking value 1 on the element ∆. Remark 1. We use the simple fact that a linear operator in W 2 can be identified with an infinite-dimensional complex-valued matrix in some basis of this space. Theorem 2. Let us consider the differential equation ih
∂Ψ = HΨ, ∂t
(1)
where Ψ ∈ W 2 , h is a real nonzero constant, (hkm )k,m∈N determined by the operator H is a diagonal complex-valued matrix. i Then the phase flow ∑ (W 2 , e− h Ht )t∈R defined by (1) preserves the ∑ measure ν∆ if and only if the series Im(hkk ) is absolutely convergent and Im(hkk ) = k∈N
k∈N
0. Proof. Let X ∈ B(W 2 ). Then ν(e− h Ht (X)) = ν(e− h H1 t × e− h H2 t (X)), i
i
i
where H 1 = (h1k,j )i,j∈N ia a diagonal matrix with h1ii = Re(hii ) for i ∈ N , and H2 = (h2ii )i∈N is also a diagonal matrix with h2ii = Im(hii ) for i ∈ N. Using Theorem 1 we have ν(e− h tH1 × e− h tH2 (X)) = ν(e− h tH2 (X)). i
i
i
Note that the matrix − hi H2 is a real-valued diagonal matrix. Finally, using the scheme considered in Theorem 9 of Section 8, we easily obtain the validity of Theorem 2. Theorem 3. Let us consider the Schr˝ odinger equation i¯h
∂Ψ = HΨ, ∂t
(2)
where H is a totally continuous Hermytian operator defined in L2 (R3 ; C), ¯h is Plank’s constant (see Section 9). Then there exists a σ-finite Borel measure µ in L2 (R3 , C) which is invariant under the group Γ of transformations of L2 (R3 , C) generated by the onei parameter group (e− h¯ tH )t∈R defined by (2) and by the group of all translations 2 3 h of L (R , C) having the form ∑ h(Ψ) = Ψ + ak Ψk , 1≤k≤n
129
where ak ∈ C and (Ψk )k∈N is the family of all proper orthonormal vectors, which corresponds to the family (λk )k∈N of all proper numbers of the operator H and generates a basis in L2 (R3 , C). Proof. Let us define the functional µ by (∀X)(X ∈ B(W 2 ) → µ({Ψ : (< Ψ, Ψk >)k∈N ∈ X}) = ν(X)). By Theorem 2, it readily follows that the functional ν satisfies all conditions of Theorem 3.
130
11. Property of Essential Uniqueness for Measures Let (E, S) be a measurable space. Let K be some class of σ-finite non-trivial measures defined on the measurable space (E, S). A measurable set X ∈ S is said to have the property of essential uniqueness with respect to the class K if (∀Y )(∀µ)(∀λ)(Y ∈ S & µ ∈ K & λ ∈ K → → µ(X ∩ Y ) = λ(X ∩ Y )). The class of all measurable subsets of the space E whose every element has the property of essential uniqueness with respect to the class K is denoted by S(K). We begin our discussion with the following proposition. Theorem 1. The class S(K) is a hereditary σ-ring (with respect to the σ-algebra S) of subsets of the space E, i.e., (∀X)(∀Y )(X ∈ S(K) & Y ∈ S & Y ⊆ X → Y ∈ S(K)).
Proof. Let X ∈ S(K). Then (∀Z)(∀Y )(Z ∈ S & Y ∈ S → (∀µ)(∀λ)(µ ∈ K & λ ∈ K → → µ(X ∩ (Z ∩ Y )) = λ(X ∩ (Z ∩ Y )))). Note that if Y ⊆ X, then µ(X ∩ (Z ∩ Y )) = µ(Y ∩ Z), λ(X ∩ (Z ∩ Y )) = λ(Y ∩ Z). In view of the relation considered above, we have (∀Z)(Z ∈ S → (∀µ)(∀λ)(µ ∈ S(K) & λ ∈ S(K) → → µ(Y ∩ Z) = λ(Y ∩ Z))). This proves the validity of the relation Y ∈ S(K). Let us show that (∀X)(∀Y )(X ∈ S(K) & Y ∈ S(K) → X ∪ Y ∈ S(K)). 131
Indeed, since (∀Z)(Z ∈ S → (∀µ)(∀λ)(µ ∈ K & λ ∈ K → → µ(Z ∩ (X \ (X ∩ Y ))) = λ(Z ∩ (X \ (X ∩ Y )))&
& µ(Z ∩(X ∩Y )) = λ(Z ∩(X ∩Y ))) & µ(Z ∩(Y \(X ∩Y ))) = λ(Z ∩(Y \(X ∩Y ))), we have µ(Z ∩ (X ∪ Y )) = λ(Z ∩ (X ∪ Y )). The latter relation implies X ∪ Y ∈ S(K). Let (Xk )k∈N be an arbitrary sequence of elements of the class S(K). Using the latter relation, we can assume that the family (Xk )k∈N is disjoint. Note that (∀k)(∀Z)(k ∈ N & Z ∈ S(K) → (∀µ)(∀λ)(µ ∈ S(K) & & λ ∈ S(K) → µ(Z ∩ Xk ) = λ(Z ∩ Xk )). Using the property of the σ-finite measures µ and λ, we can assume (∀k)(k ∈ N → 0 < µ(Xk ) < ∞). We have )) ∑ ( ( ∪ µ(Xk ∩ Z) = Xk = µ Z∩ =
∑
k∈N
k∈N
(
λ(Xk ∩ Z) = λ Z ∩
( ∪
k∈N
)) Xk
,
k∈N
which means the validity of the relation ∪ Xk ∈ S(K). k∈N
The proof is completed. Remark 1. One can easily construct an example of the measurable space (E, S) with a class K of σ-finite nontrivial measures such that S(K) is not a σ-algebra. 132
The following theorem concerns the existence of a maximal (in the sense of measure) element from the class S(K). Theorem 2. There exists an element X0 ∈ S(K) such that (∀Z)(∀µ)(Z ∈ S(K) & µ ∈ K → µ(Z \ X0 ) = 0). The set X0 is a maximal (in the sense of measure) subset of the space E which belongs to the class S(K). Proof. Let λ be an arbitrary element of the class K. Consider a probability e which is equivalent to the measure λ. measure λ Denote a=
e sup λ(X). X∈S(K)
It is clear that e n ) > a − 1 )). (∀n)(n ∈ N → (∃Xn )(Xn ∈ S(K) & λ(X n Let us consider the set ∪ X0 = Xk . k∈N
Using Theorem 1, we have X0 ∈ S(K). Let Z ∈ S(K), µ ∈ K and µ(Z \ X0 ) > 0. e \ X0 ) is also a strictly positive number. Note that It is clear that α = λ(Z (Z \ X0 ) ∈ S(K) and λ|Z\X0 = µ|Z\X0 . Therefore, we have e λ((Z \ X0 ) ∪ X0 ) = a + α > a, and we get a contradiction with the definition of the number a. Theorem 2 is proved. We now wish to present one construction closely connected with Theorem 2. Let Γ be a subgroup of the group RN such that: 1) If (g1 , g2 , · · ·) ∈ Γ, then (∀n)(n ∈ N → (g1 , · · · , gn , 0, 0, · · ·) ∈ Γ), 133
2) Γ ⊆ G[0;1]N , where G[0;1]N is a subgroup of the group RN considered in Section 6. Denote the measure ν[0;1]N by ν (see Section 6). Assume also that (∀n)(n ∈ N → An =
n ∏ i=1
Ri ×
∏
△i ),
i>n
where (∀i)(i ∈ N → Ri = R & △i = [0; 1]). The following theorem is valid. ∪ Theorem 3. The set n∈N An has the property of essential uniqueness with respect to the class K0 of all Γ-invariant σ-finite Borel measures, taking the value one on the element [0; 1]N if and only if the group Γ is everywhere dense in the space RN with respect to the usual Tikhonov topology. ∪ Proof. Assume that n∈N An has the property of essential uniqueness with respect to the class K0 of all Γ-invariant σ-finite Borel measures taking the value one on the element [0; 1]N , i.e., (∀X)(∀µ)(∀λ)(X ∈ B(RN ) & µ ∈ K0 & λ ∈ K0 → ( ( ∪ )) ( ( ∪ )) →µ X∩ An =λ X∩ An ). n∈N
Denote by
(n) Bx,r
n∈N
the open ball with center at a point x and radius r, i.e.,
n { } ∑ (n) Bx,r = y|y ∈ RN & (yi − xi )2 < r2 , i=1
where n ∈ N, x = (x1 , · · · , xn ) ∈ R . Assume that the group Γ is not everywhere dense in RN . Then, for some natural number n ∈ N , there exist a real positive number r > 0 and a point x ∈ Rn such that ( ) (n) Bx,r × RN \{1,···,n} ∩ Γ = ∅. n
∏ (n) Let us consider the set B0, r × i>n △i . 4 It is clear that ( ( ) ∏ ) (n) (n) ν B0, r × △i = bn B0,r4 > 0, 4
i>n
134
where by bn is denoted the n-dimensional standard Borel measure on the space Rn . Since the measure ν is σ-finite, there exists a countable Γ-configuration B ∏ (n) of the set B0, r × i>n △i , being ν-almost Γ-invariant. 4 Note that the condition ∏ (n) Bx, r × △i ⊆ R N B 4
i>n
is valid. Indeed, assume the contrary. Then, for some element g ∈ Γ, we have ( ∏ ) ( (n) ∏ ) (n) g B0, r × △i ∩ Bx, r × △i ̸= ∅. 4
4
i>n
i>n (n)
which means that there exists a point x e = (x1 , · · ·) ∈ B0, r × 4 the condition ∏ (n) △i g(e x) ∈ Bx,r4 ×
∏ i>n
△i such that
i>n
is valid, where g(e x) = (g1 (x1 ), g2 (x2 ), · · · , gn (xn ), · · ·). By the property of the group Γ, we have ge = (g1 , · · · , gn , 0, 0, · · ·) ∈ Γ. Also, for ge ∈ Γ, we have ( ∏ ) ( (n) ∏ ) (n) ge B0, r × △i ∩ Bx, r × △i = ∅, 4
4
i>n
i>n
from which we get ( ∏ ∏ ) (n) (n) △i ⊆ Bx,r × △i , ge B0, r × 4
i>n
i>n
i. e., (n) ge = ge(0, 0, · · ·) = (g1 , · · · , gn , 0, 0, · · ·) ∈ Bx,r × RN \{1,···,n}
which is a contradiction. The necessity is proved. Let us prove the sufficiency. Let µ be an arbitrary element of K0 . Since Γ is everywhere dense in RN , the set of all elements (g1 , g2 , · · · , gn+k ) ∈ Rn+k , where 135
(g1 , · · · , gn+k , 0, 0, · · ·) ∈ Γ, is everywhere dense in Rn+k . By the property of essential uniqueness of the standard Borel measure bn+k in the space Rn+k , we have ( ( ) ) ∏ (∀X) X ∈ B(Rn+k ) → µ X × △i = bn+k (X) . i>n+k
Hence we obtain ( ) ( ) ∏ ∏ ν X× △i = µ X × △i i>n+k
i>n+k
Using Carath´eodory theorem, we easily conclude that (∀Y )(Y ∈ B(RN ) → µ(Y ∩ An ) = ν(Y ∩ An )). The latter relation means that a subset An has the property of essential uniqueness with respect to the class K0 for arbitrary n ∈ N. An application of Theorem 1 completes the proof of the sufficiency. Theorem 3 gives rise to ∪ Theorem 4. The set n∈N An is a maximal (in the sense of measure) subset of the space RN having the property of essential uniqueness in the class K0 , i.e., ( ( ) ) ∪ (∀X)(∀µ) X ∈ S(K0 ) & µ ∈ K0 → µ X \ An = 0 . n∈N
Proof. By Theorem 3, we can easily prove that ( ( ) ) ∪ (∀X) ∀µ)(X ∈ S(K0 ) & µ ∈ K0 → µ X \ An = 0 . n∈N
Assume the contrary and let, for some set Z0 ∈ S(K0 ) and some measure µ0 ∈ K0 , ( ) ∪ µ0 Z0 \ An > 0. ∪
n∈N
It is clear that Z0 \ n∈N An ∈ S(K0 ). On the other hand, for the measure ν, we have 136
( ) ∪ ν Z0 \ An = 0, n∈N
which is a contradiction with the condition Z0 ∈ S(K0 ) since ( ) ( ) ∪ ∪ µ0 Z0 \ An ̸= ν Z0 \ An . n∈N
n∈N
Example 1. Let (E, S) be a measurable space, K be some class of σ-finite measures defined on the space (E, S). We say that a set X ⊆ E has the property of being single-valued in the class K if (∀µ)(∀λ)(µ ∈ K & λ ∈ K → µ(X) = λ(X)). It is clear that every element of the class S(K) has the property of being single-valued. On the other hand, we can easily conctruct an example of the measurable space (E, S) with a class of σ-finite nontrivial measures, for which the class of all subsets with the property of being single-valued is larger than the class S(K). Example 2. Let Γ ⊆ RN be a subgroup of RN such that (∀g)(g = (g1 , · · · , gn , · · ·) & g ∈ RN → (∀n)(n ∈ N →
∪
→ (g1 , · · · , gn , 0, 0, · · ·) ∈ Γ)).
Note that the set n∈N An has the property of essential uniqueness in the class of all σ-finite Γ-invariant Borel measures taking value 1 on the element [0; 1]N if and only if the following conditions are satisfied: 1) card(Γ/G[0;1]N ) ≤ ℵ0 ; 2) The group Γ is everywhere dense in the space RN with respect to the Tikhonov topology. Example 3. If the conditions mentioned in Example 2 are satisfied, then: ∪ a) For 2 ≤ card(Γ/G[0;1]N ) ≤ ℵ0 , the element n∈N An is not a maximal (in the sense of measure) subset of the space RN with the property of essential uniqueness in the class K0 . b) For 2 ≤ card(Γ/G[0;1]N ) ≤ ℵ0 , we can easily construct an example of a maximal (in the sense of measure) subset of the space RN with the property of essential uniqueness in the class of all Γ-invariant σ-finite Borel measures taking the value one on the element [0; 1]N .
137
12.The Structure of Elementary σ-Finite Invariant Measures In this section, some properties of the so-called elementary invariant measures are discussed. In particular, the structure of such measures is described. In this connection, let us recall some definitions from the theory of invariant measures. We say that a triple (E, G, S) is an invariant measurable space if the following conditions hold: 1) E is a nonempty set; 2) G is some group of transformations of E; 3) S is some G-invariant σ-algebra of subsets of E. Let (E, G, S, µ) be an invariant measure space. We say that a function µX : S → R+ (X ∈ S) defined by µX (Z) = µ(X ∩ Z) (Z ∈ S) is a component of the measure µ, generated by the set X. Let (E, G, S, µ) be an invariant measure space and let µ be a σ-finite invariant measure. We say that the component µX is an elementary component of the measure µ if, for arbitrary Z ∈ S with µX (Z) > 0, there exists a sequence (gk )k∈N of elements from the group G such that ( ) ∪ µX X \ gk (Z) = 0. k∈N
The next auxiliary statement is useful for many questions of the general theory of invariant measures. Lemma 1. Let (E, G, S, µ) be an invariant measure space with a σ-finite invariant measure. Then there exists a family (µXi )i∈I consisting of all elementary component of the measure µ such that: a)(∀i)(i ∈ I → µ(Xi ) > 0); b)(∀i)(∀j)(i ∈ I & j ∈ I & i ̸= j → µ(Xi ∩ Xj ) = 0); c)(∀X)(∀µX )(X ∈ S & µX is an elementary part of the measure µ → (∃i)(i ∈ I & µ(X△Xi ) = 0)).
138
(The proof of Lemma 1 can be found in [67]). Let (E, G, S, µ) be an invariant measure space and (µXi )i∈I be some family its elementary components satisfying conditions a) − c) of Lemma 1. Following [67], the measure µ is called elementary (represented) if the representation ∑ µ= µXi i∈I
is valid. Otherwise, the measure µ is called nonelementary. Let (µi )i∈I be some family of mutually orthogonal G-invariant measures defined on a G-invariant measure space (E, G, S). We say that this family is a basis of invariant measurable space (E, G, S) if, for an arbitrary elementary G-invariant measure µ, there exists a sequence of nonpositive real numbers (ai )i∈J , where J ⊆ I and card(J) ≤ ℵ0 , such that ( ) ∑ (∀Y ) Y ∈ S → µ(Y ) = ai µi (Y ) . i∈J
The following auxiliary lemma is valid. Lemma 2. Assume that µ is a G-invariant non-trivial σ-finite measure with the property of metrical transitivity. Let λ be an arbitrary G-invariant σ-finite measure which is absolutery continuous with respect to the measure µ. Then there exists a coefficient q ≥ 0 such that λ = q · µ.
Proof. By the Radon-Nikodym theorem (see Section 1), there exists a measurable function f :E→R
+
such that, for an arbitrary set Z ∈ S, we have ∫ λ(Z) = f dµ. Z
Lemma 2 will be proved if we show that the function f is µ-almost everywhere constant. Let g be an arbitrary element of the group G. Then, for Z ∈ S, we have ∫ ∫ ∫ (f og)dµ = f dµ = µ(g(Z)) = µ(Z) = f dµ. Z
g(Z)
Z
139
Since Z ∈ S is an arbitrary measurable subset of the space E, for µ-almost points of the space E, we get (f og)(x) = f (x). Now, it is clear that the function f is constant. Indeed, for arbitrary positive real numbers a and b denote by Za,b the set {x|x ∈ E & a ≤ f (x) ≤ b}. It is clear that, for g ∈ G, we have µ(g(Za,b )△Za,b ) = 0, which means that Za,b is an almost G-invariant subset of the space E. Since the measure µ has the property of metrical transitivity, the condition µ(E \ Za,b ) = 0 ∨ µ(Za,b ) = 0 is valid. Hence we can construct a sequence of segments [am ; bm ]m∈N such that: 1) (∀m)(m ∈ N → [am+1 ; bm+1 ] ⊆ [am ; bm ]); 2) limm→∞ (bm − am ) = 0; 3) (∀m)(m ∈ N → µ(E \ Zam ,bm ) = 0). If q is the common point of these segments, then the function f almost everywhere (in the sense of the measure µ) takes the value q, and Lemma 2 is proved. Lemma 3. Let µ and λ be two σ-finite G-invariant measures. Then we have λ = λ 1 + λ2 , where λ1 and λ2 are σ-finite G-invariant measures such that λ1 is absolutely continuous with respect to µ, and λ2 is orthogonal with respect to µ. ∑ Proof. Let λ = i∈I λi be represented as sum of his elementary components. It is clear that, by the properties of the measures λ, we have card(I) ≤ ℵ0 . Using Lemma 2, we can prove that (∀i)(i ∈ I → (the measure λi is orthogonal with respect to the measure µ)∨( measure λi is absolutery continuous with respect to the measure µ)). 140
Denote I0 = {i|i ∈ I & (λi is absolutely continuous with respect to the measure µ)}. Then the validity of the representation λ=
∑
λi +
i∈I0
∑
λi
i∈I\I0
is obvious. The proof is completed. Let (λk )1≤k≤n be a family of G-invariant signed measures. This family is called independent if the equality n ∑
αk λk = 0
k=1
holds if, and only if for an arbitrary integer k ∈ [1; n], we have αk = 0 (1 ≤ k ≤ n). A family (λi )i∈I of G-invariant signed measures is called ℵ0 -independent if (∀I0 )(I0 ⊆ I & Card(I0 ) ≤ ℵ0 &
∑
αi λ i = 0 →
i∈I0
∑
|αi | = 0).
i∈I0
The following theorem is valid. Theorem 1. Let (E, G, S) be an invariant measurable space and suppose that the class of all elementary G-invariant measures defined on S is not an empty set. Then there exists a family (µi )i∈I of σ-finite measures such that: 1) (∀i)(i ∈ I → (µi is a σ-finite G-invariant measure with the property of metrical transitivity )); 2) The family (µi )i∈I is ℵ0 -independent; 3) (µi )i∈I is a family of mutually orthogonal measures; 4) for an arbitrary elementary σ-finite G-invariant measure µ, there exist a countable subset I0 ⊆ I and a countable family of positive numbers (ai )i∈I0 such that µ=
∑
ai µi ,
i∈I0
and this representation is unique. Proof. Let (λξ )ξ 0, denote by nδ a natural number such that ∞ ∑ 1 < δ2. 2 (i + 1) i=n δ
Assume (∀k)(1 ≤ k ≤ nδ → hk = 0), 1 (∀k)(k ≥ nδ → hk = ). k+1 It is clear that h = (hk )k∈N ∈ / G∆ , ||h|| < δ, and ∆ ∩ (∆ + h) = ∅. Theorem 5 is proved. Summarizing all the above results, we obtain the following statement. Theorem 6. The duality between the measure λ and the Baire category with respect to the sentence P0 , where P0 = ((∀X)(X ⊆ ℓ2 & Xis a Baire subset of second category → (∃δ)(δ > 0 → (∀x)(||x|| < δ → X ∩ (X + x) ̸= ∅)))), is not valid. Remark 3. By Remark 2 and Theorem 4, one easily obtains the validity of the duality between the linear Lebesgue measure and the Baire category with respect to the sentence P0 in R (see e.g. [67]). Remark 4. Using Theorem 6, we conclude that an analogy of the Erd˝osSierpi´ nski Duality Principle is not valid for the measure λ and the Baire category in the infinite-dimensional separable Hilbert Space ℓ2 . There are also several important works devoted to the solution of analogous problems in various topological vector spaces (see, e.g., [46], [47], [67] and others). The following notion is frequently useful in studying various questions of measure theory. We say that the measure µ defined in a topological vector space (E, T ) satisfies the axiom of Steinhaus if the condition (∀X)(X ∈ dom(µ) & µ(X) < ∞ → (∀ϵ)(ϵ > 0 → ((there exists a neighbourhood 147
Vϵ of the neitral element ) & ((∀h)(h ∈ Vϵ → µ(h(X)△X) < ϵ)))) holds. Theorem 7. The measure λ does not satisfy the axiom of Steinhaus. Proof. Assume the contrary. Then for the set ∆ and for the number ϵ = 12 , there exists a number δ > 0 such that ((∀x)(||x|| < δ → λ((∆ + x) △ ∆)
0 → (∃δ)(δ > 0 → (∀h)(||h|| < δ → (X+h)∩X ̸= ∅))). This means that the duality between the measure µ0 (which is not σ-finite) and the Baire category with respect to the property P0 is valid in the separable Hilbert space ℓ2 . Also note that the measure µ0 satisfies the axiom of Steinhaus In connection with the above results, one can pose a problem of the validity of the duality between the translate-invariant Borel measure (constructed in section 6) and the Baire category with respect to the property P0 in the infinitedimensional separable Hilbert space ℓ2 .
148
14. A Formula of Substituting Variables in the Integral for Invariant Measures in the Separable Hilbert Space ℓ2 Let Rn (n ≥ 1) be the n-dimensional Euclidean space. Denote by C (1) the class of all one-to-one continuously differentiable transformations of the space Rn onto itself. Let bn be the n-dimensional standard Borel measure defined on the space Rn . The function Y(U ) defined by ∂U k (Y(U ))(x1 , · · · , xn ) = , ∂ξp 1≤p≤n1≤k≤n where U (ξ1 , · · · , ξn ) = (U1 (ξ1 , · · · , ξn ), · · · , Un (ξ1 , · · · , ξn )) and U ∈ C (1) , is called the Jacobian of the transformation U . It is well-known that if D is a compact subset in the space Rn and f is a bounded real-valued continuous function on Rn , then the substitution formula is valid in an n-multiple integral having the form ∫ ∫ f (ξ1 , · · · , ξn )dbn (ξ1 , · · · , ξn ) = f (U (ξ1 , · · · , ξn )× U −1 (D)
D
×(Y(U ))(ξ1 , · · · , ξn ) × d(bn (ξ1 , · · · , ξn )). Let l2 be the standard Hilbert space and A be a linear operator in this space for which we have ( (∀(xk )k∈N ) (xk )k∈N ∈ l2 → A(x1 , · · · , xn , · · ·) =
a11 a12 · · · a1m · · · a21 a22 .. .. . .
= am1 .. .
x1 x2 .. .
xm amm .. .. . .
) ,
where A(ek ) = (ak1 , ak2 , · · ·) for ek = (0, · · · , 0, 1, 0, · · ·) (k ≤ 1). | {z } k−1
For an arbitrary natural number n ∈ N , denote by An an n × n matrix defined by a11 · · · a1n .. . An = ... . an1 · · · ann 149
A real number a ∈ R is called the determinant of the operator A if a = lim det(An ). n→∞
Let U : l2 → l2 be an one-to-one continuously differentiable transformation of the space l2 such that (∀(xk )k∈N )((xk )k∈N ∈ l2 → U ((xk )k∈N ) = (U1 ((xk )k∈N ), · · ·)). The determinant of an infinite matrix (if it exists) ( ∂U ) ∂U (ξ1 , · · ·) k = det ∂(x1 , · · ·) ∂ξp p∈N k∈N is called the Jacobian Y(U ) of the transformation U , where (ξk )k∈N ∈ l2 . The following question is the object of interest. Problem. Does there exist a nontrivial σ-finite Borel measure µ in the Hilbert space l2 such that, for an arbitrary one-to-one continuously differentiable transformation on the Hilbert space l2 , the following condition is valid: (∀f )(∀B)(f is a bounded real-valued continuous function on the space l2 & B ∈ ∫ ∫ B(l2 )) → B f dµ = U −1 (B) |Y(U )|dµ))? That is why it is interesting to consider the question about the existence of a measure µ in the Hilbert space l2 which is an analogue of the Lebesgue measure and for which the formula of substitution of variables in the integral is valid. An answer to the above question is contained in the following assertion. Theorem 1. For an arbitary nontrivial σ-finite Borel measure µ, there exist a Borel subset B ⊆ l2 , a one-to-one continuously defferentiable transformation U of the space l2 and a bounded continuous function f defined on the space l2 such that ∫ ∫ f dµ ̸= f (U )|Y(U )|dµ. B
U −1 (B)
Proof. Let µ be an arbitrary nontrivial σ-finite Borel measure. Since µ is not l2 -invariant, there exist a subset B ∈ B(l2 ) and a translation g ∈ l2 such that µ(B + g) ̸= µ(B). 150
Let us put (∀x)(x ∈ l2 → U (x) = x + g & f (x) = 1). It is clear that |Y(U )| = 1. Assume the contrary, i.e. suppose that ∫ ∫ f dµ = f (u)|Y(U )|dµ. U −1 (B)
B
Then ∫
∫ dµ =
dµ. U −1 (B)
B
Notice that ∫ µ(B) =
∫
dµ = µ(U −1 (B)).
dµ = U −1 (B)
B
Thus, we have a contradiction with the condition µ(B + g) ̸= µ(B). The theorem is proved. Recall the notation △=
∏[
0;
i∈N
1 ] 2i+1
.
Let ν△ be the measure constructed in Section 6. Since the group of all translations of the space l2 is a proper subgroup of the group of all mutually one-to-one continuously defferentiable transformations of the space l2 , it is of interest to consider the problem of characterizing the maximal group Γ△ of all translations of the space l2 for which the formula of substitution of variables in the integral is valid for the measure ν△ . We have the following assertion. Theorem 2. {h|(∀B)(∀f )(((B ∈ B(l2 )) & (f is bounded continous function defined on l2 ) → ∫ ∫ f dν△ = f (x + h) × |Y(Uh )|(x)dν△ (x)} = B
B−h
151
= {h|(∀B)(B ∈ B(l2 ) → ν△ (B + h) = ν△ (B))}, where Uh (x) = x + h (x ∈ ℓ2 ), i.e., Γ△ = {(Ck )k∈N |(Ck )k∈N ∈ l2 & (∃n(Ck )k∈N ) ∞ ∑
(n(Ck )k∈N ∈ N → the series
( ln 1 −
p=n(Ck )k∈N
|Cp | ) is convergent)}. bp − ap
Proof. The validity of the inclusion Γ△ ⊆ G△ is obvious. Let us show the validity of the inverse inclusion. Assume h ∈ G△ . To prove the relation h ∈ Γ△ , it is sufficient to show (∀A)(∀B)(A ∈ B(l2 ) & B ∈ B(l2 ) → ∫ →
∫ IA (x + h) × |Y(Uh )|(x)dν△ (x)).
IA (x)dν△ (x) = B
B−h
But the latter statement follows from the G△ -invariance of the measure µ△ , since ∫ IA dν△ = ν△ (A ∩ B) = B
∫
= ν△ ((A − h) ∩ (B − h)) =
IA−h (x)dν△ (x) = B−h
∫
IA (x + h) × |Y(Uh )|(x)dν△ (x).
= B−h
Using Theorem 1 of Section 6, we complete the proof of Theorem 2.
152
15. The Zero-One Law for Invariant Measures The following definition is important for the theory of invariant measures and its various applications. Let (E, S, µ) be a measurable space with measure µ. Let G be a group of transformations of the space E such that (∀h)(∀B)(h ∈ G & B ∈ S → h(B) ∈ S). We say that a set B0 ∈ S with µ(B0 ) > 0 satisfies the zero-one law with respect to the pair (µ, G) if ( ) µ(g(B0 )) µ(g(B0 )) (∀g) g ∈ G → =1∨ =0 . µ(B0 ) µ(B0 ) The zero-one law is realized in the following situation with the infinitedimensional Hilbert space ℓ2 . Theorem 1. Let △0 =
∏[ 1 ] 0; i+1 . 2
i∈N
Then an arbitrary element B ∈ B(l2 ) ∩ (∪n∈N An ) with µ∆ (B) > 0, where ( ∏ [ 1 ]) (∀n) n ∈ N → An = Rn × 0; i+1 , 2 i≥n+1
satisfies the zero-one law with respect to the pair (ν△ , l2 ). Proof. Note that it is sufficient to show the validity of the relation ( ( ∪ ) (∀B)(∀h) B ∈ B(l2 ) ∩ An & ν△0 (B) > 0 & n∈N
) &h ∈ l2 \ G△0 → µ(B + h) = 0 . Since h = (h1 , h2 , · · ·) ∈ l2 \ G△0 , either for any natural number k ∈ N the series ∞ ∑
ln(1 − |hn | × 2n+1 )
n=k
∑∞ has no sense or for some natural number k0 the series n=k0 ln(1 − |hn | × 2n+1 ) is divergent. In the first case, there exists a subsequence (np )p∈N of N such that 153
1 − |hnp | × 2np +1 < 0, i.e, |hnp | >
1 . 2np +1
The second case means that ) ( ∪ (∏[ ) 1 ] An = ∅ 0; i+1 + h ∩ 2 i∈N
n∈N
from which we have ( ∪
) ( ∪ ) An + h ∩ An = ∅.
n∈N
n∈N
In particular, (B + h) ∩
( ∪
) An = ∅.
n∈N
Consider the second case where, for some k0 ∈ N , the series ∑
ln(1 − |hn | × 2n+1 )
n≥k0
is divergent. Two subcases are possible. I. ∑ We can choose a subsequence (np )p≥1 of natural numbers such that: ∞ np +1 ) = −∞, a) p=1 ln(1 − |hnp | × 2 np +1 b) 0 < 1 − 2 × |hnp | < 1. II.∑There exists a subsequence (nm )m≥1 of natural numbers such that: ∞ nm +1 ) = +∞, c) m=1 ln(1 − |hnm | × 2 nm +1 d) 1 < 1 − |hnm | × 2 . Note that II cannot take place since |hnm | × 2nm +1 ≥ 0. Assume that condition I holds. This means that there exists a subsequence (npl )l∈N of (np )p∈N such that hnpl+1 ≤ 2 × hnpl ; hence we have ν△0
(( ∪
) ( ∪ )) An + h ∩ An = ∅.
n∈N
n∈N
154
In particular, ( ( ∪ ) ( ( ∪ )) ) (∀B) B ∈ B(l2 ) ∩ An → ν△0 (B + h) ∩ An =0 . n∈N
n∈N
Since ( ) ( ( ∪ )) ∪ ν△0 (B + h) = ν△0 (B + h) \ An + ν△0 (B + h) ∩ An , n∈N
n∈N
we obtain ν△0 (B + h) = 0, which completes the proof of Theorem 1. Remark 1. Note that not all elements B ∈ B(l2 ) satisfy the zero-one law with respect to the pair (ν△0 , l2 ). Indeed, if we consider the element B0 ∈ B(l2 ) defined by ( ) 1 1 1 B0 = △0 ∪ (△0 + 2, 1, 1, , , · · · , , · · · ∪ 2 3 n ( ) 1 1 1 ∪(△0 + 1, 2, 1, , , · · · , , · · · , 2 3 n then, on the one hand, ν△ (B0 ) = 1 ( ) and, on the other hand, for h = − 1, 1, 1, 12 , 13 , · · · , n1 , · · · we have ν△0 (B0 + h) = 2, ν△0 (B0 ) which means that the set B0 does not satisfy the zero-one law with respect to the pair (ν△0 , l2 ). It will be interesting to describe the class of all Borel subsets of the space l2 which satisfy the zero-one law with respect to the pair (ν△0 , l2 ).
155
16. Absolutely Negligible and Absolutely Nonmeasurable Sets in the Topological Vector Space RN It is well-known that, in the Euclidean space Rn , nonmeasurable (in the sense of the Lebesgue measure) subsets preserve some properties under translations. Among such properties are, for example, the Baire property, nonmeasurability in the sense of Lebesgue, the Vitali property, Bernstein’s property and so on. It is the object of interest to consider analogous questions in the infinite dimensional topological vector space RN . The following notions are important for our further investigation of certain sets in the infinite-dimensional topological vector space RN . Recall that a set X ⊆ RN has the Baire property if there exist sets G ⊆ RN , X1 ⊆ RN and X2 ⊆ RN such that X = (G \ X1 ) ∪ X2 , where G is an open set in RN , X1 and X2 are some subsets of RN of first category. A set Y ⊆ RN is called a Bernstein set if (∀F )(F ⊆ RN & F is closed in R & card(F ) = c → F ∩ Y ̸= ∅ & F ∩ (RN \ Y ) ̸= ∅). N
A set Z ⊆ RN is called a Vitali set if there exist a vector subspace G ⊆ RN and a G-invariant σ-finite non-trivial Borel measure µ such that for some µmeasurable set X with µ(X) > 0 and for some countable subgroup G0 ⊆ G the following conditions are satisfied: a) a set Y intersects every set of the factor space G/G0 at most in one point; b) Y ⊆ X & X ⊆ ∪g∈G0 g(Y ). We have the following Theorem 1. The following conditions hold: a) (∀X)(∀h)(X ⊆ RN & h ∈ RN & (X has the property of Baire) → (X + h has the property of Baire)); b) (∀X)(∀h)(X ⊆ RN & h ∈ RN & (X is a Bernstein set) → (X + h is a Bernstein set)). The proof of Theorem 1 is easy one. Theorem 2. Let µ be an arbitrary nontrivial G-invariant (G ⊆ RN ) Borel measure and let a group G consists of a freely acting uncountable subgroup of 156
the additive group RN . If we denote by L the completion of the class B(RN ) by the measure µ, then (∃h0 )(∃Y )(h0 ∈ RN & Y is a Vitali set & Y ∈ L → Y + h0 ̸∈ L). The proof of Theorem 2 is based on the following lemmas. Lemma 1. Assume that (E, G, S, µ) is a measurable space with a nontrivial σ-finite G-invariant measure. Let a group G contain an uncountable freely acting subgroup. Then every set X with µ(X) > 0 contains a µ-nonmeasurable subset which is the Vitali set. Lemma 2. A RN -quasiinvariant Borel probability measure defined on the space RN does not exist. The proof of Lemmas 1 and 2 can be found in [68]. Proof of Theorem 2. Let us consider a probability measure µ0 which is equivalent to the measure µ. Using Lemma 2, we have (∃B0 )(∃h0 )(B0 ∈ B(RN ) & µ0 (B0 ) > 0 & h0 ∈ RN → µ0 (B0 + h0 ) = 0). Since the measure µ0 is equivalent to the measure µ, we get µ(B0 ) > 0 &
µ(B0 + h0 ) = 0.
It is clear that, for the completion overlineµ of the measure µ, we have µ(B0 ) > 0 & µ(B0 + h0 ) = 0. By Lemma 1, we conclude that there exists a µ-nonmeasurable Vitali set Y ⊆ B0 . On the other hand, since Y + h0 ⊆ B0 + h0 and µ is a completion of the measure µ, we obtain Y + h0 ∈ L & Y ̸∈ L. Q.E.D. Remark 1. Theorem 2 states that the σ-algebra of subsets of the space RN obtained by the operation of completion (with respect to a certain σ-finite invariant Borel measure) is not an RN -invariant σ-algebra. An analogue of this proposition is not true in the finite-dimensional Euclidean space Rn (It is sufficient to consider the Lebesgue measure ln (n ≥ 1)).
157
Remark 2. An analogue of Theorem 2 is not valid in the Euclidean space Rn . Theorem 2 is not true either for Bernstein subsets (see [98].) Summarizing the results of Theorems 1 and 2, we conclude that the nonmeasurability property of Vitali subsets is not preserved under translations, but the property of absolute nonmeasurability of the Bernstein set is preserved in that case. Let us return the following important notion from the theory of groups. Let (E, G, S) be an invariant measurable space. We recall that a group G of transformations acts freely in E if (∀x)(∀g)(x ∈ E & g ∈ G &g ̸= IdE → g(x) ̸= x). The following statement shows us that the nonmeasurability of Vitaly sets is not preserved under transformations from a given group. Theorem 3. Assume that (E, G, S, µ) is a space with a σ-finite G-invariant measure. Let the group G contains an uncountable freely acting subgroup and G0 be a group of transformations of the space E with respect to which the measure µ is not G0 -quasiinvariant. Then there exists a Vitali set Y such that: a) Y ̸∈ domµ; b) (∃h0 )(h0 ∈ G0 → h0 + Y ∈ dom(µ)). The proof of Theorem 3 is based on one classical result of A.B. Kharazishvili stating that, in such situations, every µ-measurable set with a positive measure µ contains a Vitali nonmeasurable subset (see [68]). The following lemma is of some interest. Lemma 3. Let µ be an arbitrary σ-finite nontrivial Borel measure defined on the uncountable Polish space. If we denote by µ the completion of the measure µ, then the following relation is valid: (∀X)(∀Y )(X is a Bernstein set & µ(Y ) > 0 → X ∩ Y ̸∈ dom(µ)).
One can easily get the proof of this lemma. The main property of Bernstein sets in infinite-dimensional vector space RN is presented in the next theorem.
158
Theorem 4. Let µ be a nonatomic σ-finite Borel measure defined on the space RN . Assume X to be an arbitrary Bernstein set. Then there exists a subset X0 ⊆ X such that a) X0 ̸∈ dom(µ), b) (∃h0 )(h0 ∈ RN → X0 + h0 ∈ dom(µ)), where µ denotes the completion of the measure µ. Proof. Without loss of generality, we may assume that µ is a probability measure. By using one result∪ of Ulam, we have that the measure µ is concentrated on a countable union n∈N Kn of compact subsets of the space RN (see Section 18). Since RN is not covered by the countable union of compact subsets ∪ N K such that n , there exists a translation h0 ∈ R n∈N ( ∪ ) ( ∪ ) Kn ∩ Kn + h0 = ∅. n∈N
n∈N
N Let ∪ X be an arbitrary Bernstein subset of the space R . Consider X0 = X ∩ ( n∈N Kn ). By Lemma 3, we have
X0 ̸∈ dom(µ). ∪ On the other hand, since X0 + h0 ⊆ RN \ n∈N Kn , by the property of the measure µ, we obtain X0 + h0 ∈ dom(µ). Remark 3. Using the Bernstein construction, we can obtain a generalization of Theorem 2 whose proof is based on the Vitali construction. Namely, on the one hand, we prove that the domain of completion of an arbitrary σ-finite nontrivial Borel measure µ is not RN -invariant. On the other hand, this means that an arbitrary nonatomic nontrivial σfinite Borel measure defined on the space RN is not complete. Remark 4. An analogue of Theorem 3 is not true for the Euclidean space Rn (n ≥ 1). Let E be a base space, G be a group of transformations of E and let X be a subset of the space E. X is called a G-absolutely negligible set if for any G-invariant σ-finite measure µ, there exists its G- invariant extension µ e such that X ∈ dom(e µ and µ e(X) = 0.
159
A geometrical characterization of absolutely negligible subsets due to A.B. Kharazishvili, is presented in the next theorem. Theorem 5. Let E be a base space, G be a group of transformations of E containing some uncountable subgroup acting freely in E, and X be an arbitrary subset of the space E. Then the following two conditions are equivalent: 1) X is a G-absolutely negligible subset of the space E; 2) for an arbitrary countable G-configuration X ′ of the set X, there exists a countable sequence (gk )k∈N of elements of G such that ∩
gk (X ′ ) = ∅.
k∈N
For the proof of Theorem 5, see [67]. Now, we can formulate and prove the following statement. Theorem 6. Let E1 be a nonempty base space, G1 be a group of transformations of E1 , containing an uncountable subgroup acting freely in E1 , and X1 be a G1 -absolutely negligible subset of the space E1 . Further, let E2 be an arbitrary nonempty set and G2 be an arbitrary group of transformations of E2 . Then for any subset X2 ⊆ E2 the set X1 × X2 is a G1 × G2 -absolutely negligible subset of the space E1 × E2 . Proof. Note that a group G1 × G2 satisfies the conditions formulated in (1) (2) Theorem 1. Indeed, let (e gk )k∈N = (gk , gk )k∈N be an arbitrary countable subgroup of the group G1 × G2 . Consider X′ =
∪
(1)
gk (X1 ).
k∈N (1)
By Theorem 5, we conclude that there exists a countable subgroup (hp )p∈N such that ( ∪ ) ∩ (1) h(1) g (X ) = ∅. 1 p k p∈N
k∈N
Let h be an arbitrary element of the group G2 . Then it is easy to prove that ∪
(1)
(2)
(gk , gk )(X1 × X2 ) ⊆
k∈N
160
⊆
∪
(1)
(2)
(gk , gk )(X1 × X2 ) ⊆
k∈N
⊆
∪
(1)
gk (X1 ) × E2 .
k∈N
On the other hand, we have ( ∪ ) ∩ (1) (2) (h(1) (gk , gk )(X1 × X2 ) ⊆ p , h) p∈N
⊆
k∈N
∩ (
h(1) p
( ∪
p∈N
(1)
gk (X1 )
))
× E2 .
k∈N
Using the condition ∩
h(1) p
( ∪
p∈N
) (1) gk (X1 ) = ∅,
k∈N
we obtain ∩ p∈N
( ∪ ) (1) (2) (h(1) (gk , gk )(X1 × X2 ) ⊆ ∅. p , h) k∈N
By Theorem 5, we conclude that the set X1 × X2 is a G1 × G2 -absolutely negligible subset of the space E1 × E2 , and Theorem 2 is proved. We will need the following important result established by A.B. Kharazishvili. Theorem 7. Let Γ be an arbitrary uncountable subgroup of the additive group R. Then there exists a countable family (Xn )n∈N of subsets of R such that: 1) (∀n)(n ∪ ∈ N → ( the set Xn is a Γ-absolutely negligible subset of R); 2) R = n∈N Xn . For the proof of Theorem 7, see [68]. The following proposition is valid. Theorem 8. Let RN be the space of all sequences of real numbers, R(N ) be the group of all finite sequences of the space RN , i.e., R(N ) = {(xk )k∈N |(xk )k∈N ∈ RN & Card{k : xk ̸= 0} < ℵ0 }. Then there exists a countable family (Zn )n≥1 of subsets of RN such that: 161
1) (∀n)(n∪∈ N → ( the set Zn is Γ-absolutely negligible subset of RN )). 2) RN = n∈N Zn . The proof of Theorem 8 will be obtained if we use Theorem 6 and the obvious equality R(N ) = R × RN \{0} . Let E be a base space and let Γ be a group of its transformations. A set Y ⊆ E is said to be Γ-absolutely nonmeasurable if, for every nontrivial σ-finite Γ-invariant (Γ-quasiinvariant) measure µ defined on E, we have Y ̸∈ dom(µ). One simple method of the construction of absolutely nonmeasurable subsets in product spaces is presented in the following theorem. Theorem 9. Let E1 be a base space, Γ1 be a group of transformations of E1 , Y1 be a Γ1 -absolutery nonmeasurable subset of the space E1 . Further, let E2 be also a base space and Γ2 be a group of its transformations. Then a subset Y1 × E2 of the space E1 × E2 is Γ1 × Γ2 -absolutely nonmeasurable. Proof. Let µ be a Γ1 × Γ2 -quasiinvariant probability measure defined on E1 × E2 such that Y1 × E2 ∈ dom(µ). It is clear that the class F defined by the formula F = {A : A ∈ dom(µ) & A = X × E2 } is a Γ1 ×Γ2 -invariant σ-algebra of subsets of the space E1 ×E2 and F ⊆ dom(µ). Let us consider the class F1 defined by F1 = {X : X ⊆ E1 & X × E2 ∈ F }. It is also clear that the class F1 is a Γ1 -invariant σ-algebra of subsets of the space E1 . The functional ψ defined by (∀X)(X ∈ F1 → ψ(X) = µ(X × E2 )) is a Γ1 -invariant probability measure such that Y1 ∈ dom(ψ). This contradicts the Γ1 -absolute nonmeasurability of Y1 , and Theorem 5 is proved. In the sequel, we will need the following result obtained in [67].
162
Theorem 10. In the Euclidean space Rn there exists an Rn -absolutely nonmeasurable subset. By Theorems 9 and 10, we can obtain the validity of the following theorem. Theorem 11. In the infinite-dimensional vector space RN there exists an R -absolutely nonmeasurable subset. (N )
Example 1. Let X ∗ be an Rn -absolutely nonmeasurable subset of Rn and X ∗∗ be R(N \{1,···,n}) -absolutely negligible subset; then X ∗ × X ∗∗ is not an R(N ) -absolutely nonmeasurable subset because it is an R(N ) -absolutely negligible subset of RN . Remark 5. The results of Theorems 7 and 11 formulated for the group R(N ) are also true for the group G△ ⊆ RN considered in Section 6. Remark 6. The proofs of Theorems 7 and 11 are largely based on the results obtained in [46].
163
17. Independent Families of Sets and Some of Their Applications in Measure Theory Some methods of the combinatorial set theory have lately been successfully used in different areas of mathematics, for example, in topology and measure theory. Among of them, special mention should be made of the method of constructing a maximal (in the sense of cardinality) family of independent families of sets in arbitrary infinite base spaces. The question of the existence of a maximal (in the sense of cardinality) independent family of subsets was considered by A. Tarski(see e.g.[66]). He proved that this cardinality is equal to 2card(E) . This result found an interesting application in general topology by means of which it was proved that in an arbitrary infinite space E the cardinality of the card(E) class of all ultrafilters is equal to 22 (see e.g.[66]). Using the method of an independent family in the case of the Euclidean space En , A.B. Kharazishvili constructed a maximal (in the sense of cardinality) family of orthogonal elementary Dn -invariant extensions of the Lebesgue measure (see[67]). E. Szpilrajn (E. Marczewski) was the first who suggested the method of constructing nonseparable extensions of the Lebesgue measure (see [127]). Later, S. Kakutani, K. Kodaira and J. Oxtoby constructed nonseparable invariant extensions of the Lebesgue measure (see [61], [80]). We must say that the abovementioned methods can be successfully generalized to some classes of topological groups (see e.g. [45]). In [67], the method of an independent family of sets is used to construct an example of a nonelementary Dn -invariant extension µ of the Lebesgue measure ln such that the topological weight a(µ) of the metric space (dom(µ), ρµ ) associated with the measure µ is maximal; in particular, this cardinality is equal to the cardinal number 2c , where c denotes the cardinality of the continuum. In this section, the method of an independent family of sets due to A.Tarski, is generalized and successfully used to construct a maximal family of orthogonal invariant (elementary and nonelementary) extensions of the Haar measure defined on an arbitrary locally-compact σ-compact topological group. One method of construction of nonelementary invariant extensions of the Haar measure is considered, by means of which one problem formulated in [67] is solved. Let us recall some definitions which are used in this section. Let E be the main base space and β be some cardinal number. We say that a family (Xi )i∈I of subsets of the set E is β-independent if the condition ( ) ∩ (∀J) J ⊂ I & card(J) < β → X i ̸= ∅ , i∈J
holds, where ( ) (∀i)(i ∈ I → (X i = Xi ) ∨ X i = (E\Xi )) . 164
We say that a family (Xi )i∈I of subsets of the space E is strictly β-independent if the condition ∩
(∀J)(J ⊂ I & card(J) ≤ β →
X i ̸= ∅)
i∈J
holds, where (∀i)(i ∈ I → (X i = Xi ) ∨ (X i = E\Xi )). Remark 1. It is clear that the β-independence of the family (Xi )i∈I does not imply its strictly β-independence. The question of the existence of an ℵ0 -independent family of subsets of an arbitrary infinite space E, with maximal cardinality, was solved by A. Tarski. card(E) He proved that this power is equal to 22 . This result gave rise to many interesting applications, in particular, it was proved in general topology that the cardinality of all ultrafilters defined in an arbitrary infinite space E is equal card(E) to 22 . If the cardinality of the base space E is equal to the continuum, one can prove the existence of a strictly ℵ0 -independent family of subsets of E with a maximal possible cardinality 2c , where c is the cardinality of the continuum. This result found applications in the theory of extensions of the Lebesgue measure, in particular, it was used to construct Dn -invariant extensions of the Lebesgue measure. Let us consider the problem of the existence of a maximal (in the sense of cardinality) strictly independent family of subsets in an arbitrary infinite space. The next auxiliary proposition plays the key role in this section. Theorem 1. If an infinite set E satisfies the condition Card(E β ) = card(E), where β is an infinite cardinal number, then there exists a maximal (in the sense of cardinality) strictly β-independent family (Xi )i∈I of subsets of the space E, such that card(I) = 2card(E) .
Proof. Let us prove the theorem in four steps. 165
I. Assume that X is a set such that card(X) = card(E). Let (Xi )i∈I be a partition of the set X such that card(I) = card(E), ( ) (∀i) i ∈ I → card(Xi ) = card(E) . Let us consider the class Ω′ of subsets of X defined by { } Ω′ = Y | Y ⊆ X & (∀i)(i ∈ I → card(Y ∩ Xi ) = 1) . The class Ω′ satisfies the conditions card(Ω′ ) = card(B(E)), ( ) (∀Y )(∀Z) Y ∈ Ω′ & Z ∈ Ω′ & Y ̸= Z → Y \ Z ̸= ∅ . II. Assume that (Xi )i∈I is a partition of X such that card(I) = card(E), ( ) (∀i) i ∈ I → card(Xi ) = card(E) . For an arbitrary index i ∈ I, denote by Ω′i a class of all subsets of Xi such that card(Ω′i ) = card(B(E)), ) ( (∀Y )(∀Z) Y ∈ Ω′i & Z ∈ Ω′i & Y ̸= Z → Y \ Z ̸= ∅ . Fix the set J with card(J) = card(B(E)) and consider a representation (Yij )j∈J of the class Ω′i such that (∀j)(j ∈ J → Yij ∈ Ω′i ). Let us put
{ } ∪ Ω′′ = Y |(∃j)(j ∈ J & Y = Yij . i∈I ′′
It is easy to verify that the class Ω satisfies the condition card(Ω′′ ) = card(B(E)), ( (∀Y )(∀Z) Y ∈ Ω′′ & Z ∈ Ω′′ & Y ̸= Z → 166
) → card(Y \ Z) = card(E) . III. Denote by U the class of subsets of Y defined by U = {Z|(Y ∈ Ω′′ & Z is class of all β-powerful subsets of the set Y } Assume that (Zξ )0≤ξ≤ωβ is an arbitrary sequence of elements of U . Let us prove that ( ) ∪ card Z0 \ Zξ ≥ card(E). ξ∈]0;ωβ [
Let (Yξ )0≤ξ≤ωβ be a sequence from the class Ω′′ which corresponds to the sequence (Zξ )0≤ξ≤ωβ . Since the power of the class Ω′′ is equal to 2card(E) , the sequence (Yξ )0≤ξ≤ωβ can be continued as for the sequence (Yξ )ξ 0. By Lemma 7, we obtain the existence of an index i0 ∈ [0; 1] and a point x0 ∈ R such that {i0 } × {xi0 } ∈ (A \ B) ∩ ({i0 } × Xi0 ). This means that xi0 ∈ ψ −1 (A) \ ψ −1 (B). 187
We have obtained a contradiction with the condition ψ −1 (A) = ψ −1 (B) and the correctness of the definition of the functional µ is proved. Let us verify that µ is a D1 -invariant functional. It is sufficient to verify the validity of the condition (∀a)(∀b)(∀c)(∀d)(0 ≤ a < b ≤ 1 & − ∞ < c < d < ∞ & g ∈ D1 → → µ(g ◦ ψ −1 ([a; b] × [c; d])) = µ(ψ −1 ([a; b] × [c; d]))). Note that µ(g ◦ ψ −1 ([a; b] × [c; d])) = µ(ψ −1 ([a; b] × g([c; d]))). Indeed, by Lemma 7, on the one hand, g ◦ ψ −1 ([a; b] × [c; d]) = g((
∪
Xi ) ∩ [c; d]) =
i∈[a;b]
= ((
∪
Xi ) ∩ g([c; d])) \ X ′ ∪ X ′′ ,
i∈[a;b]
where X ′ and X ′′ are some subsets of the space R whose cardinalities are strictly less than c, and, on the other hand, ∪ ψ −1 ([a; b] × [c; d]) = ( Xi ) ∩ g([c; d]). i∈[a;b]
By the inclusion g ◦ ψ −1 ([a; b] × [c; d])△ψ −1 ([a; b] × g([c; d])) ⊆ X ′ ∪ X ′′ , we have µ(g ◦ ψ −1 ([a; b] × [c; d])) = µ(ψ −1 ([a; b] × g([c; d]))). e 1 -invariance of the measure λ we have Finally, by the D µ(g ◦ ψ −1 ([a; b] × [c; d])) = µ(ψ −1 ([a; b] × g([c; d]))) = = λ([a; b] × g([c; d])) = λ([a; b] × [c; d]) = µ(ψ −1 ([a; b] × [c; d])). The nonelementarity of the measure µ can be established by the scheme considered in the proof of Theorem 5. Theorem 8 is proved. The basic result is formulated by the following theorem. 188
Theorem 9. There exists a µ-measurable subset of the space R which has only one density point with respect to the Vitali standard system generated by the family of all open intervals of R. Proof. Let us consider a µ-measurable set ψ −1 (K), where K is the closed square whose vertices are at the points (0; 0), ( 21 , 21 ), (1; 0), ( 12 , −1 2 ). Let x ∈ R and {(ak , bk )}k∈N be an arbitrary sequence of open intervals, fundamental at the point x. By using the construction of the measure µ, we have 1 − 2|x|, |x| < 12 −1 µ(ψ (K) ∩ (ak ; bk )) lim = . k→∞ µ((ak ; bk )) 0, |x| ≥ 12 In particular, this means that the set ψ −1 (K) has only one density point (x = 0) with respect to the standard Vitali system generated by the family of all open intervals of the real line R. Remark 3. Applying the scheme proposed in Theorems 8 and 9, we can construct a Dn -invariant nonelementary extension µn of the classical Lebesque measure ln in the Euclidean space Rn , for n ≥ 2, such that some µn -measurable set would have only one density point with respect to the sdandard Vitali system generated by the family of all n-dimensional cubes of the space Rn , and the topological weight of the metric space (dom(µn ), ϱµn ) would be maximal, in particular, be equal to 2c . Remark 4. We must say that stronger (than Theorem 9) results were obtained in [74].
189
18. Separated Families of Probability Measures Here we need some notions and auxilary propositions from probability theory. Let (E, S) be a measurable space. Definition 1. A family of probability measures (µi )i∈I defined on a measurable space (E, S) is called weakly separated if there exists a family (Xi )i∈I of measurable subsets of E, such that (∀i)(i ∈ I & j ∈ I → µi (Xj ) = δ(i, j)), where δ(i, j) denotes Kronecker’s function defined on the Cartesian square I 2 of the set I. Definition 2. A family of probability measures (µi )i∈I defined on a measurable space (E, S) is called strictly separated if there exists a disjoint family (Xi )i∈I of measurable subsets of the space E, such that (∀i)(i ∈ I → µi (Xi ) = 1). It is clear that an arbitrary strictly separated family (µi )i∈I of probability measures is weakly separated. In general statistical analysis there often arises a question of transition from a weakly separated family of probability measures to the corresponding strictly separated family. In this context, the following result is the object of interest Theorem 1. In the system of axioms (ZF C) the following conditions are equivalent: 1) The Continuum Hypothesis is true; 2) an arbitrary weakly separated family of probability measures is strictly separated. Proof. 1) → 2). Let (µξ )ξ≺ω1 be a continuous family of probability measures defined on a measurable space (E, S) and suppose that there exists a family (Xξ )ξ≺ω1 of measurable subsets of the space E such that (∀ξ)(∀τ )(ξ ≺ ω1 & τ ≺ ω1 → µξ (Xτ ) = δ(ξ, τ )), where δ(ξ, τ ) denotes Kronecker’s function defined on the Cartesian square [0; ω1 [×[0; ω1 [ of the space [0; ω1 [. Let ( ) ∪ (∀ξ) ξ ≺ ω1 → Yξ = Xξ \ Xτ . τ ≺ξ
190
It is clear that (Yξ )ξ≺ω1 is a disjoint family of measurable subsets of the space E, such that (∀ξ)(ξ ≺ ω1 → µξ (Yξ ) = 1). This means that the relation 1) → 2) is proved. 2) → 1). For arbitrary x ∈]0; 1[, define the σ-algebra Bx of subsets of the space △2 =]0; 1[×]0; 1[ by Bx = {Y |Y ⊆ △2 & (card(Y ∩ ({x}×]0; 1[)) ≤ ℵ0 )∨ (card({x}×]0; 1[\Y ) ≤ ℵ0 )}. For arbitrary x ∈]1; 2[, denote by Bx the σ-algebra of subsets of the space △2 defined by Bx = {Y |Y ⊆ △2 & (card(Y ∩ (]0; 1[×{x − 1})) ≤ ℵ0 ) ∨ . (card(]0; 1[×{x − 1} \ Y ) ≤ ℵ0 )}. Assume ∩
S=
Bx
x∈]0;1[∪]1;2[
. It is clear that every element of the families ({x}×]0; 1[)x∈]0;1[ and (]0; 1[×{x − 1})x∈]1;2[ belongs to the σ-algebra S. Define the family (µt )t∈]0;1[∪]1;2[ of probability measures by {
1, if card(({t}×]0; 1[) \ Z) ≤ ℵ0 , ) , 0, if card(({t}×]0; 1[) ∩ Z) ≤ ℵ0
{
1, if card((]0; 1[×{t − 1}) \ Z) ≤ ℵ0 , ) . 0, if card((]0; 1[×{t − 1}) ∩ Z) ≤ ℵ0
(∀t)(t ∈]0; 1[→ (∀Z)(Z ∈ S → µt (Z) =
(∀t)(t ∈]1; 2[→ (∀Z)(Z ∈ S → µt (Z) =
Let us consider the family (Xt )t∈]0;1[∪]1;2[ of measurable subsets of the space △2 , where { {t}×]0; 1[, if t < 1 ) (∀t)(t ∈]0; 1[∪]1; 2[→ Xt = . ]0; 1[×{t − 1}, if t > 1 191
It is clear that the family (µt )t∈]0;1[∪]1;2[ of probability measures is weakly separated because of ( )2 (∀t1 )(∀t2 )((t1 , t2 ) ∈ ]0; 1[∪]1; 2[ → µt1 (Xt2 ) = δ(t1 , t2 )), where δ(., .) denotes Kronecker’s function defined on the Cartesian square (]0; 1[∪]1; 2[)2 of the set ]0; 1[∪]1; 2[. From condition 1) we have that the family (µt )t∈]0;1[∪]1;2[ of probability measures is strictly separated. This means that there exists a family of disjoint measurable subsets (Yt )t∈]0;1[∪]1;2[ such that (∀t)(t ∈]0; 1[∪]1; 2[→ µt (Yt ) = 1). ∪ ∪ Let us consider the set A = t∈]0;1[ Yt and B = t∈]1;2[ Yt . It is clear that A and B do not intersect each other. On the other hand, we have (∀x)(x ∈]0; 1[→ card(({x}×]0; 1[) ∩ B) ≤ ℵ0 & & card((]0; 1[×{x}) ∩ A) ≤ ℵ0 ). Denote by (Cξ )ξ≺ω1 some injective transfinite sequence of horizontal segments of the space △2 . It is clear that )) ( ( ∪ Cξ ≤ ℵ0 × ℵ0 = ℵ0 . card A ∩ ξ≺ω1
(∪ ) We must prove that the ortogonal projection of the set A ∩ C ξ≺ω1 ξ on the interval ]0; 1[×{0} coincides with this interval. Indeed, let a be an arbitrary vertical segment of the space △2 . Since card(B∩ a) ≤ ℵ0 , there exists an ordinal index ξ0 ≺ ω1 such that the point(of intersection ) ∪ of Cξ0 and a belongs to the set A. This means that the set A ∩ ξ≺ω1 Cξ is projected on the whole interval ]0; 1[×{0} and,therefore, 2ℵ0 ≤ ℵ1 . The proof is completed. Remark 1. In the system of axioms (ZF C)&(¬CH)&(M A) the family of probability measures (µt )t∈]0;1[∪]1;2[ considered in Theorem 1 is an example of weakly separated probability measures which is not strictly separated. Remark 2. It is interesting to note that the pair {A, B} constructed in Theorem 1 coincides with the Sierpinski partition of the unit square ]0; 1[2 ( see, e.g.[67]). 192
Let us consider the question of transition from a weakly separated family of probability measures to the strictly separated one, when the family of probability measures is defined on a metric space. Lemma 1. Let (E, ρ) be a metric space. Then, for an arbitrary probability Borel measure µ defined on the Borel σ -algebra B(E) of subsets of E, the following condition (∀B)(∀ϵ)(B ∈ B(E) & ϵ > 0 → (∃F )(F is closed in E → µ(B \ F ) < ϵ)) holds. Proof. We put A(E) = {B : B ∈ B(E) & (∀ϵ)(ϵ > 0 → (∃F )(F ⊂ B & F is closed in E → µ(B \ F ) < ϵ))}. Let us verify that the class A(E) is a σ-algebra. Let (Bk )k∈N be a countable family of elements of A(E). We fix a number ϵ > 0 and choose a family (Fk )k∈N of closed subsets of E, such that Fi ⊂ Bi & µ(Bi \ Fi )
j Ujk for arbitrary i ∈ I. Clearly, the family of subsets (Zik )i∈I is disjoint and the union of its elements covers the space E. Remark that the subset Zik can be represented in the form k Zik = ∪m∈N Zim , k where Zim is closed in E for all k ∈ I and k ∈ N . The cardinality of the set of all indexes i ∈ I, for which µ(Zik ) > 0, is countable because the functional µ is a probability measure. Let us denote this set of indexes by T (k), i.e.,
T (k) = {i : i ∈ I & µ(Zik ) > 0}. We put J(k) = J \ T (k). Let us verify that µ(∪i∈J(k) Zik ) = 0. Indeed, if we assume the contrary µ(∪i∈J(k) Zik ) > 0, 195
then, from the validity of the representation k ∪i∈J Zik = ∪m∈N (∪i∈J Zim ) k for an arbitrary part J ⊆ J(k) and using Lemma 1, we conclude that ∪i∈J Zim is a Borel subset of E. Hence, if we define
(∀J)(J ⊆ J(k) → λ(J) = µ(∪i∈J Zik )), then the measurability in a wide sense of the parameter set J(k) is obvious, because λ is a nontrivial finite measure on J(k) and it takes value zero at all points of J(k). This is a contradiction, and we conclude that the probability measure µ is concentrated on the set U (k) = ∪i∈T (k) Uik ⊃ ∪i∈T (k) Zik . Clearly, the intersection ∩k≥1 U (k) is a separable subset of E and its µ-measure is equal to 1. Finally, we can put that E(µ) is a closure of ∩k≥1 U (k), and Lemma 4 is proved. According to the property of separable closed subsets in a complete metric space, we have the validity of the following proposition. Lemma 5. An arbitrary complete metric space (E, ρ) whose topological weight is not measurable in a wide sense is a Radon space. The following example demonstrates the fact that the class of all Radon metric spaces is wider then the class of all Polish spaces. Example 1. Let ω1 be the first uncountable cardinal number. We define ∑ F = {(xi )i∈ω1 : (xi )i∈ω1 ∈ Rω1 & x2i < +∞}, i∈ω1
(∀(xi )i∈ω1 )(∀(yi )i∈ω1 )((xi )i∈ω1 ∈ F & (yi )i∈ω1 ∈ F → √∑ → ρ((xi )i∈ω1 , (yi )i∈ω1 ) = (xi − yi )2 . i∈ω1
Note that (F, ρ) is an example of a complete metric space(actually, Hilbert space), whose topological weight is equal to ω1 . On the one hand, it is clear that (F, ρ) is not a Polish space because it is not separable. On the other hand, using the well-known Ulam’s result about the nonmeasurability in a wide sense of the cardinal number ω1 (see, e.g. [67]), we conclude that (F, ρ) is a metric 196
space whose topological weight is not measurable in a wide sense. According to Lemma 5, we conclude that (F, ρ) is a Radon space. The following important result is essentially due to Martin and Solovay. Lemma 6. Let (F, ρ) be a complete separable metric space equipped with some probability Borel measure µ. If (Ei )i∈I is a family of µ-zero measurable subsets of F , such that card(I) ≤ c, then in the ∪system of axioms (ZF C) & (M A) the outer measure µ∗ of the union E = i∈I Ei is equal to zero. Proof. Let ϵ > 0. We must find an open set including E which has µmeasure ≤ ϵ. Let P be the collection of all open sets of measure < ϵ; and for p, q ∈ P let p ≼ q mean p ⊆ q. We first show that the set P satisfies the c.c.c. Let Q be a pairwise incompatible subset of P . Let Qn = {p ∈ Q : m(p) ≤ (1 − 21n ϵ)}. It is enough to show that Ωn is countable. Let E ′ ⊂ E be a countable set everywhere dense in E. For each p ∈ Qn , choose p ⊆ p so that p is a finite union of open balls with center at E ′ and rational radius such that µ(p − p) < 2ϵn . Since all such finite unions constitute only a countable family, it suffices to show that if p and q are distinct members of Qn , then p ̸= q. Suppose p = q. Then p ∪ q ⊆ (p \ p) ∪ q, so µ(p ∪ q) < 2ϵn + (1 − 21n )ϵ = ϵ. This implies that p and q are compatible, which contradicts p, q ∈ Ω. For i ∈ I, let Di = {p ∈ P : Ei ⊆ p}. Using the relation µ(Ei ) = 0, we easily conclude that Di is dense in P . By Martin’s axiom it follows that there is a subnet Q of P which meets every Di . Let G be the union of the members of Q. Then G is open. From Di ∩ Q ̸= ∅ we find that Ei ⊆ G, so E ⊆ G. It remains to show that µ(G) > ϵ leads to a contradiction. By Lindel˝of’s theorem, G is a countable union of sets in Q. It follows that there is a finite union G1 of sets in Q such that µ(G1 ) > ϵ. But since Q is a subnet, some member of Q includes G1 and hence has measure ≥ ϵ. This is a contradiction, and Lemma 6 is proved. The following theorem is valid. Theorem 2. Let (F, ρ) be a complete metric space whose topological weight is not measurable in a wide sense. Let (µi )i∈I be a weakly separated family of Borel measures with card(I) ≤ 2ℵ0 defined on the metric space (F, ρ). 197
In the system of axioms (ZF C) & (M A), the family (µi )i∈I is strictly separated. Proof. Note that an arbitrary probability Borel measure µ, defined on the metric space (F, ρ) has the property (∀J)(∀(Xi )i∈J )((card(J) ≺ 2ℵ0 & (∀i)(i ∈ J → → µ(Xi ) = 0) → µ∗
(∪
) Xi = 0).
i∈J
Indeed, by Lemma 5, for µ there exists a separable closed support F (µ) in (F, ρ). Let us consider the equality [( ∪ ) ] [ ( ∪ )] ∪ Xi = Xi ∩ F (µ) ∪ (F \ F (µ)) ∩ Xi . i∈J
i∈J
(∪
)
i∈J
∗ Using Lemma 6, we conclude that the set i∈J Xi ∩ F (µ) is a µ -measure zero subset ) of the set F (µ). Note that the outer measure of the set (F \ F (µ)) ∩ ( ∪ i∈J Xi is equal to zero because µ(F \ F (µ)) = 0.
Let (µi )i∈J be a weakly separated family of Borel probability measures with card(J) ≤ 2ℵ0 . Let us present this family by an injective sequence (µξ )ξ≺ωα , where the first ordinal number of cardinality J is denoted by ωα . Since the family (µξ )ξ≺ωα is weakly separated, there exists a family (Xξ )ξ≺ωα of Borel subsets of the metric space F , such that (∀ξ)(∀τ )(ξ ∈ [0; ωα [ & τ ∈ [0; ωα [→ µξ (Xτ ) = δ(ξ, τ )), where σ(., .) denotes Kronecker’s function defined on the Cartesian square [0; ωα [2 of the set [0; ωα [. Let us define an ωα -sequence of subsets (Bξ )ξ≺ωα of the metric space F , such that: 1) 2) 3) 4)
(∀ξ)(ξ ≺ ωα → Bξ is a Borel subset in F ); (∀ξ)(ξ ≺ ωα → Bξ ⊆ Xξ ); (∀τ1 )(∀τ2 )(τ1 ≺ ωα & τ2 ≺ ωα & τ1 ̸= τ2 → Bτ1 ∩ Bτ2 = ∅); (∀τ )(τ ≺ ωα → µτ (Bτ ) = 1).
Assume B0 = X0 . Let, for ξ ≺ ωα , the partial sequence (Bτ )τ ≺ξ be already constructed. It is clear that (∪ ) µ∗ξ Bτ = 0. τ ≺ξ
This means that there exists a Borel subset Yξ of the space F , such that 198
∪
Bτ ⊆ Yξ and µξ (Yξ ) = 0.
τ ≺ξ
We put Bξ = Xξ \ Yξ . Now, the ωα -sequence (Bξ )ξ≺ωα of disjoint measurable subsets of the space F is constructed so that (∀ξ)(ξ ≺ ωα → µξ (Bξ ) = 1). This completes the proof of the theorem. Remark 3. Theorem 2 generalizes one result of Z. S. Zerakidze obtained in [134].
199
Appendix §1. Proof of the Michelsky-Sverchkovsky Theorem We will consider some important results and facts from measure theory which have various interesting applications to the theory of invariant and quasiinvariant measures in infinite-dimensional vector spaces. The following notion of a specific kind of partial ordering frequently turns out useful in studying various questions of measure theory and general topology. Definition 1. A partial ordering (T, ≼) is called a tree if T has the least element and, for each y ∈ T the set {x ∈ T : x ≼ y} is well-ordered by ≼. The least element of T is called the root of T . For any ordinal number α the α-th level of T is the set Tα = {y : {x ∈ T : x ≺ y} has order type α}. The height of a tree is a least ordinal α such that the α-th level of T is empty. Let A be a non-empty set and let α be an ordinal. A complete A-ary tree of height α, which consists of all functions from ∪β≺α Aβ and is ordered by inclusion, is denoted by A≺α . If A = {0; 1}, then complete A-ary trees are called binary trees. Any linearly ordered subset of the tree (T, ≼) is called a branch in T . A subset P of T is called a path through the tree T if P is a branch and contains exactly one element from each nonempty level of T . The following fundamental statement was established by K˝onig. Theorem 1. Suppose that (T, ≼) is a tree of height ω such that all levels of T are finite. Then there exists a path through T . Proof. Let x0 be the root of T . For each n ∈ ω \ {0}, we can recursively pick an element xn ∈ Tn such that xn ≻ xn−1 and the set {y ∈ T : xn ≺ y} is finite. This is possible since every level Tn of T is infinite. Then (xn )n∈N is a path through T . Q.E.D. Let us recall some notions which are necessary to formulate the Axiom of Determinacy.
200
An arbitrary subset A ⊂ ω ω determines a game of type GA between two players denoted by I and II. A game of type GA is described as follows: Player I writes a natural number a0 . His opponent player II, knowing the number a0 , writes a number a1 . Player I, knowing the number a1 and remembing his number a0 , writes his new number a2 . Futher, player II, looking at a0 , a1 , a2 , writes a number a3 , and so on. In this infinite game, the sequence of natural numbers α = (a0 , a1 , · · · , ) is obtained. If this summarizing sequence belongs to the set A, then player I wins a game of type GA . Otherwise, player II wins this game in other cases. Various kind of games (for example, chess, checkers and so on) can be represented by the above scheme. Let ω (ω) is be the set of all finite sequences of natural numbers, including an epmty set. The last set can be considered as a sequence of length zero. Definition 2. A function σ : ω (ω) → ω is called a strategy in the game of type GA . Let σ be a strategy of player I and τ be a strategy of player II. Acording to the strategies σ and τ , we have a0 = σ(∅), a1 = τ (a0 ) = τ (σ(∅)), a2 = σ(a0 , a1 ) = σ(σ(∅), τ (σ(∅))), · · · . Definition 3. A function σ : ω (ω) → ω is called a winning strategy for player I if (a0 , a1 , a2 , · · ·) ∈ A for an arbitrary strategy τ of player II. Analogously, a function τ : ω (ω) → ω is called a winning strategy for the player II if (a0 , a1 , a2 , · · ·) ∈ /A for an arbitrary strategy σ of player I. Definition 4. An infinite game of type GA is called determined if there exists a winning strategy for at least one of two players in this game. Axiom of Determinacy (denoted by AD) is the assertion that every infinite game of type GA (for arbitrary A ⊆ ω ω ) is determined.
201
We begin our discussion with the classical theorem of Michelsky and Sverchkovsky. This theorem concerns the Lebesgue measurability of all linear subsets. Theorem 2. If the Axiom of Determinacy is valid, then every subset of the real axis R is Lebesgue measurable. Proof. To prove Theorem 2 it is sufficient to show that every subset X of the real axis R has measure zero or contains a measurable subset with a positive Lebesgue measure. It can be assumed, withous loss of generality, that X ⊆ [0; 1]. The collection S of all subsets of [0; 1] represented by a finite union of intervals, half-intervals or segments with rational end points is countable. Let Wα be an element from the class S with a numerical number α. Let us fix a number ϵ > 0. We say that an infinite sequence of natural numbers α = (a0 , a1 , · · ·) is ϵ-correct if the following two conditions hold: 1) every number a2n is equal to 0 or 1; 2) The Lebesgue measure of the subset Wa2m+1 is less than ϵ × 2−2m for all m ∈ ω. To every such ϵ-correct α-sequence there correspond a number F (α) from the interval [0; 1] with a binary representation 0, a0 a2 a4 a6 · · · and a set Vα = Wa1 ∪ Wa3 ∪ Wa5 ∪ · · · Let A(ϵ) be the set of sequences α = (a0 , a1 , · · ·) satisfying at least one of the following two conditions: a) a sequence α is ϵ-correct and a point F (α) belongs to the difference X \Vα , b) a sequence α is not ϵ-correct and the smallest m, for which at least one of conditions 1) or 2) is not valid, does not satisfy condition 2), but satisfies condition 1). According to the AD, two cases are possible. Case I. There exists a winning strategy τϵ for player II in a game of type GA(ϵ) , where ϵ is a fixed positive number. In that case the set X has measure zero. Indeed, for the fixed number ϵ, we can construct a covering of the set X by a countable union of elements of S, whose Lebesgue measure is not greater than ϵ. Let us denote by Cm the collection of all sequences with length m + 1 consisting of 0 or 1 for all natural number m. Let U = (a0 , a2 , a4 , · · · , a2m ) ∈ Cm . Denote the conjugate sequence of numbers a1 , a3 , · · · , a2m+1 by a2i+1 = τϵ (a0 , a1 , a2 , · · · , a2i−1 , a2i ), 202
where i ∈ ω (the definition of such a sequence is possible by using mathematical recursion). Since every number a2i with i ≤ m is equal to 0 or 1, and the function τϵ is a winning strategy for player II in a game of type GA(ϵ) , the Lebesgue measure of the set Wa2i+1 must be less than or equal to ϵ × 2−2i for arbitrary i ∈ ω. In particular, the Lebesgue measure of the set W (u) = Wa2m+1 ϵ is not greater than 22m . Hence the union Z = ∪m∈ω ∪u∈Cm W (u) has a Lebesgue measure no greater than ∞ ∑ ∑
ϵ × 2−2m = ϵ
m=0 u∈Cm
∞ ∑
2m+1 2−2m = 4ϵ.
m=0
Finally, we need verify that X ⊆ Z. Let x ∈ X. Assuming that X ⊆ [0; 1], the number X can be represented by x = 0, a0 a2 a4 · · · , where every symbol a2i is equal to 0 or 1. Using mathematical recursion, let us construct a sequence a1 , a3 , a5 , · · · of numbers of type a2i+1 by a2i+1 = τϵ (a0 , a1 , · · · , a2i ), where i ∈ ω. Using the property of the winning strategy τϵ of player II in a game of type GA(ϵ) , we conclude that α ∈ / A(ϵ). Note that all numbers a2i , being the leads of player I, are equal to 0 or 1. On the one hand, using the definition of the set A(ϵ), we conclude that the sequence α is ϵ-correct and F (α) ∈ / X \ Vα . On the other hand, the point F (α) coincides with the point x ∈ X and x ∈ Vα . By the definition of the set Z, we have Vα ⊆ Z, and, correspondingly, x ∈ Z. As x was an arbitrary point of the set X, the inclusion X ⊆ Z is proved. Case 2. For a fixed positive number ϵ, there exists a winning strategy σϵ for player I in a game of type GA(ϵ) . Let m ∈ ω. Denote by ωm the collection of all indexes a ∈ ω, which correspond to the set Wa whose measure is not greater ϵ than 22m . Let us fix the numeration ωm = {a[m, l] : l ∈ ω} by natural numbers l for the whole set ωm . To every infinite sequence γ = (l0 , l1 , l2 , · · ·) of natural numbers lm let there correspond the following subjects: 203
the sequence a1 , a3 , a5 , · · · of numbers defined by a2i+1 = a[i, li ]; the sequence of numbers a0 , a2 , a4 , · · · defined by induction (with respect to i) using the mathematical recursion formula a2i = σϵ (a0 , a1 , a2 · · · , a2i−2 , a2i−1 ), where i ∈ ω is arbitrary (in particular, a0 = σϵ (∅)); the resulting sequence α(γ) = (a0 , a1 , · · ·); finally, the point H(γ) = F (α(γ)) (i.e., the point 0, a0 a2 a4 · · · in binary representation). The sequence of numbers a1 , a3 , a5 , · · · arising in this construction obviously satisfies condition 2) in the definition of an ϵ-correct sequence for any m ∈ ω. This means that by choosing σϵ as a winning strategy, the resulting sequence α(γ) (equal to σϵ∗ (a1 , a3 , a5 , · · ·)) must be ϵ-correct, while the point H(γ) must be an element of the difference X \ Vα(γ) . Hence the function H maps the Baire space N of all infinite sequences of natural numbers into the set X so that H(γ) ∈ X \ Vα(γ) for any γ ∈ N. We are going to prove that the set P = H(N ) contains a measurable subset of positive Lebesgue measure. Let us begin by verifying that the upper measure of the set P is greater than ϵ. Assume the contrary. Then the set P can covered by the countable union W = ∪m∈ω W m of pairwise nonintersecting elements W m of the family S and the total measure of these elements is not greater than ϵ. Using the validity of ∞ ∑
ϵ × 2−2m =
m=0
4 ϵ > ϵ, 3
it can be assumed, withous loss of generality, that the measure of each set W m is not greater than ϵ × 2−2m . Under this assumption, there exists a sequence γ = (l0 , l1 , l2 , · · ·) satisfying the equality W m = Wa[m,lm ] for all m. Now, on virtue of the above arguments, we have H(γ) ∈ P \ Vα(γ) . Therefore, H(γ) ∈ P \ W. But this contradicts the choice of W and the inclusion P ⊆ W is proved. Thus, the upper measure P is actually greater than ϵ. Note that the set P coincides with the union of the countable family of sets P (k) = {H(γ) : γ = (l0 , l1 , l2 , · · ·) ∈ N and l0 ≤ k}, which generates an ⊆-increasing sequence. So there exists a number k0 such that the set P0 = P (k0 ) has a Lebesgue measure greater than ϵ. Analogously, there exists a natural number k1 such that the set P1 = {H(γ) : γ = (l0 , l1 , l2 , · · ·) ∈ P0 and l1 ≤ k1 } 204
has a measure greater than ϵ. If we continue this process, then we can construct a sequence of natural numbers k0 , k1 , k2 , · · · such that the set Pm = {H(γ) : γ ∈ N & γ(i) ≤ ki for all i ≤ m} has a Lebesgue measure greater than ϵ for any m ∈ ω. Assuming the extreme left term to be equal to zero, we denote by γ(i) the i-th term li of the infinite sequence γ = (l0 , l1 , · · · , li , · · ·). We introduce the family of sets: Ym = {y ∈ [0; 1] : there exists x ∈ Pm such that the first m symbols of the binary representations of x and y coincide with deficiency }. Every set of Ym with its point y also contains some half-interval having the form [p, q[ containing the point y and having a nonzero length for y ̸= 1 (this half-interval is generated by the points of the unit interval whose first m symbols of the binary representation are fixed.) Hence all Ym are Borel sets and ,therefore, measurable. The intersection Y = ∩m∈ω Ym of all these sets is thus also measurable. Let us estimate the measure of the set Y . Since the set Ym is measurable and consists of the set Pm (it is clear from the definition of the set Ym ), the Lebesgue measure of Ym is greater than ϵ. By the construction of the sets Pm and Pm+1 , we obtain Pm+1 ⊆ Pm for all m ∈ ω, and therefore Ym+1 ⊆ Ym . So Y has a Lebesgue measure greater than ϵ. It remains to verify Y ⊆ P . Let y ∈ Y. To each natural number m there corresponds a sequence γ = (lm0 , lm1 , lm2 , · · ·) ∈ N such that the inequality lmi ≤ ki is fulfilled for i ≤ m and the first m binary symbols of numbers x and H(γm ) coincide. The family T of all finite sequences γm ⌈i = (lm0 , lm1 , · · · , lm,i−1 ) where m ∈ ω and i ≤ m, forms an infinite tree of height ω, such that all branches of T are finite. (The number of branches at each vertex of the tree is limited by the corresponding value km ). Hence, by using the K¨onig theorem, the tree T has an infinite path; in other words, there exists an infinite sequence γ = (l0 , · · ·) of natural numbers li such that the finite sequence γ⌈i = (l0 , · · · , li−1 ) belongs to the tree T for all i ∈ ω. By the definition of T , we can choose an index m(i) ≥ i for arbitrary i such that the equality γm(i) ⌈i = γ⌈i will be fulfilled. 205
By the definition of the function H, the latter equality implies the coincidence of the first i binary symbols of the real numbers H(γ) and H(γm(i) ). At the same time, by virtue of the inequality m(i) ≥ i and the choice of the sequence γm , the first i binary symbols of the numbers H(γm(i) ) and y coincide. Tending i to infinity, we obtain y = H(γ), i.e., y ∈ P. Thus, the set P (respectively, the set X since P ⊆ X) contains a measurable subset Y of positive Lebesgue measure in Case 2. This completes the proof of the Michelsky-Sverchkovsky theorem. The following result is a simple consequence of Theorem 2. Theorem 3. The Axiom of Determinacy contradicts the Axiom of Choice. Proof. On the one hand, by the Axiom of Choice (in particular, by the Vitali theorem based on an uncountable form of the (A C), we can easily construct such a subset X of the real axis R that is not measurable in the Lebesgue sense. On the other hand, using the Michelsky-Sverchkovsky theorem, we have that every subset of real axis (including X) must be measurable in the Lebesgue sense. This is a contradiction, and Theorem 3 is proved.
206
§2. A Formula of Substituting Variables into the Lebesgue Multiple Integral and Liouville’s Theorem We will consider several propositions closely connected with Liouville’s theorem. A formula of substituting variables in the Lebesgue multiple integral will be presented below. First, we need formulate a precise definition of differentiable functions on the Euclidean space Rn . Definition 1. A real-valued function f : Rn → R is called differentiable at a point x = (x1 , · · · , xn ) if there exist a family of real numbers A1 , · · · , An and a real-valued function h in Rn such that f (x1 + ∆x1 , · · · , xn + ∆xn ) − f (x1 , · · · , xn ) = =
n ∑
Ai ∆xi + h(∆x1 , · · · , ∆xn ),
i=1
where
v u n u∑ h(∆x1 , · · · , ∆xn ) lim = 0, ρ = t (∆xi )2 . ρ→0 ρ i=1
Note that if the function f is differentiable, then its coefficients Ai (1 ≤ i ≤ n) have an interesting interpretation; in particular, simple equality Ai = ∂ ∂ ∂xi f (x1 , · · · , xn ) holds for 1 ≤ q ≤ n, where ∂xi f denotes the partial derivative with respect to the variable xi . Hence ∆f (x1 , · · · , xn ) =
n ∑ ∂ f (x1 , · · · , xn )∆xi + h(∆x1 , · · · , ∆xn ). ∂x i i=1
Definition 2. A function f : Rn → R is called differentiable if it is differentiable at an arbitrary point in Rn . Definition 3. A transformation U = (U1 , · · · , Un ) : Rn → Rn is called differentiable if its all components are differentiable. Definition 4. Let U = (U1 , ·, Un ) be a differentiable transformation. The i determinant of the matrix ( ∂U ∂xj )1≤i,j≤n is called the Jacobian of the transformation U and is denoted by I(U ). The following geometrical proof of one classical result known as the formula of substituting variables in the Lebesgue multiple integral is essentially due to Ostrogradsky.
207
Theorem 1. Let U : Rn → Rn be a differentiable transformation. f : Rn → R be a real-valued Lebesgue integrable function. Then the formula ∫ ∫ (∀D)(D ∈ B(Rn ) → f dln = I(U )f (U )dln ) (10) D
U (D)
is valid. Proof. Let us consider the parallelepiped DX =
n ∏
[xi ; xi + ∆xi ]
i=1
generated by the family of vectors (Pi )1≤i≤n at the point x = (x1 , ·, xn ), where Pi = Ai Bi , Ai = (x1 , · · · , xn ), Bi = (x1 , · · · , xi−1 , xi + ∆xi , xi+1 , · · · , xn ). Under the the transformation U the vector Pi transforms into the vector U (Pi ) = U (Ai )U (Bi ). If we ignore the numbers of first order with respect to dxi (1 ≤ i ≤ n), then the Lebesgue measure of U (DX ) is equal to the Lebesgue measure of the parallelepiped constructed on the family of vectors (U (Ai )U (bi ))1≤i≤n . In our assumptions, U (Ai )U (Bi ) = (
∂Ui ∆xj )1≤j≤n (1 ≤ i ≤ n). ∂xj
Hence det((
∂Ui ∂Ui ∆xj )1≤i,j≤n ) = det(( )1≤i,j≤n × (∆ij )1≤i,j≤n )) = ∂xj ∂xj
= det((
∂Ui )1≤i,j≤n ) · det((∆ij )1≤i,j≤n ) = I(U ) · ln (∆), ∂xj {
where ∆ij =
∆i , 1,
if i = j, if i = ̸ j.
For D = ∆X , we obtain the validity of the formula (1) when the function f = 1. Note that this relation can be easily extended to the class of all Lebesgue measurable subsets D ⊆ Rn and Lebesgue intagrable functions f : Rn → R. Q.E.D. 208
Now, we will easily get the proof of Liouville’s theorem. Let us consider the system of first order linear differential equations dy = A(x)y, dx
(2)
∑ dyi = aij yj , i = 1, 2, · · · , n. dx i=1
(3)
i.e., n
(j)
Let ((yi (x))1≤i≤n )1≤j≤n be a fundamental system of solutions of system (2). Then the function W (x) defined by (j)
W (x) = det((yi (x))1≤i,j≤n ) is called the Wronskian determinant of system (2). The following proposition is valid. Theorem 1 (Liouville). The Wronskian determinant of system (2) satisfies the condition ∫x
T r(A(τ ))dτ
W (x) = W (x0 )ex0 n ∑
where T r(A(ξ)) =
,
aii (ξ), ξ, x0 , x ∈ R.
i=1
Proof. Using the well-known of differentiability of determinants, we obtain (1) (2) (n) dy1 dy1 dy1 · · · dx dx dx (1) (2) (n) · · · y2 y2 W ′ (x) = y2 = ··· · · · · · · · · · (1) (2) (n) yn yn · · · yn (1) y 1(1) dy2 = dx ··· (1) yn
(2)
···
dy2 dx
··· ··· (n) · · · yn
(2)
··· (2) yn
(1) y 1 (n) (1) y2 y dx + · · · + 2 ··· ··· y(1) n (n)
y1
y1
dy
dx
y1 (2) y2 ···
(2)
··· ··· ···
y1 (n) y2 ···
(n)
(2) dyn dx
···
(n) dyn dx
.
(k)
i If we replace the values dx standing in the right side of the above equality by the values defined by (3) for 1 ≤ i, j ≤ n, and use the property of a
209
determinant stating that the determinant does not change when we add to the elements of some row (column) the linear combination of elements of other rows (columns), then we obtain a y (1) 11 1 y2(1) ′ W (x) = ··· yn(1)
(2)
a11 y1 (2) y2 ··· (2) yn
(n)
· · · a11 y1 (n) ··· y2 ··· ··· (n) ··· yn
i.e., W ′ (x) =
n ∑
(1) y1 (1) y2 +· · ·+ ··· ann yn(1) yn(1)
(2)
y1 (2) y2 ··· (2) ann yn
aii (x)W (x).
i=1
The integration of the above equality completes the proof of Liuoville’s theorem.
210
(n)
··· y1 (n) ··· y2 ··· ··· (n) · · · ann yn
,
§3. The General Marginal Problem In this paragraph, we discuss the general marginal problem whose classical formulation is as follows. Let S be a σ-algebra of subsets of the basic space E, let qγ be a measurable mapping from the measurable space (E, S) into a probability space (Tγ , Bγ , Pγ ) for arbitrary γ ∈ Γ. Find necessary or sufficient conditions for the existence of a probability measure P on (E, S) such that Pγ = qγ (P ) for arbitrary γ ∈ Γ(see [49]). We need formulate some notions which are important for our further investigation. Let ((E, Si , Pi ))i∈I be a family of probability spaces. Definition 1. A family of probability measures (Pi )i∈I is called consistent in a wide sense if the condition (∀i)(∀j)(∀X)(i ∈ I & j ∈ I & X ∈ Si ∩ Sj → Pi (X) = Pj (X)). holds. Definition 2. A family of probabality measures (Pi )i∈I is called consistent in a narrow sense if (∀(Ak )k∈N )((∀k)(k ∈ N → Ak ∈ ∪i∈I Si ) & A0 =
∑ k∈N \{0}
∑
Ak → Pi0 (A0 ) =
k∈N \{0}
where Pik denotes an arbitrary probability measure such that the condition Ak ∈ Sik holds. 1. Note that every consistent in narrow sense family of probability measures also is consistent in wide sense. Definition 3. A family of probability measures (Pi )i∈I is called regular if, on the σ-algebra of subsets of the space E generated by the family of σ-algebras (Si )i∈I , we can define a probability measure Pσ such that (∀i )(i ∈ I → Pσ |Si = Pi ). 211
Pik (Ak )),
The probability measure Pσ is called the measure generated by a given family of probability measures (Pi )i∈I . Remark 2. It is easy to verify that every regular family of probability measures is consistent in a narrow sense. Definition 4. We say that a probability measure λ defined on a measurable space (E, S1 ) possesses a strong extension from the σ-algebra S1 onto a σalgebra S2 (S1 ⊂ S2 ) of subsets of the space E if there exists an extension µ of the measure λ from the σ-algebra S1 to the σ-algebra S2 , and , for an arbitrary σ-algebra S satisfying the condition S1 ⊂ S ⊂ S2 , and, for an arbitrary extension λ of the measure λ from the σ−algebra S1 onto the σ-algebra S, we have µ|S = λ. Let us consider some examples. Example 1. An example of a consistent in a wide sense family of probability measures (Pi )i∈I which is not consistent in a narrow sense. Let πn be a group of all translations of the Euclidean space Rn . Let P be an arbitrary πn −quasiinvariant probability Borel measure defined on the space E = Rn (n ≥ 1). Let (Xk )k∈N be a disjoint family of πn -absolutely negligible subsets of E such that ∪ Xk = E. k∈N
(The existence of such a family (Xk )k∈N is proved in [67].) Using the method of mathemathical induction we can construct a family of πn -quasiinvariant probability measures (Pk )k∈N such that: 1) (∀k)(k ∈ N → Xk ∈ dom(Pk )), 2) (∀k)(k ∈ N → dom(Pk ) ⊆ dom(Pk+1 )), 3) (∀k)(k ≥ 1 → Pk |dom(Pk−1 ) = Pk−1 ). Indeed, let us denote a measure P0 by (∀X0′ )(∀X0′′ )(∀Y0 )((X0′ is a countable πn − configuration of X0 )& &(X0′′ is a countable πn − configuration of X0 ) & (Y0 ∈ dom(P )) → 212
→ P0 ((Y0 \ X0′ ) ∪ X0′′ ) = P0 (Y0 )). It is easy to verify that P0 is a πn -quasiinvariant probability measure such that X0 ∈ dom(P0 ). Suppose that a partial family (Pi )0≤i≤k of πn -quasiinvariant probability measures is constructed such that conditions 1) − 3) hold. Let us define a probability πn -quasiinvariant measure Pk+1 by ′ ′′ ′ ′′ (∀Xk+1 )(∀Xk+1 )(∀Yk+1 )(Xk+1 and Xk+1 are countable πn −invariant
configurations of the set Xk+1 ) & (Yk+1 ∈ dom(Pk ) → ′ ′′ → µk+1 ((Yk+1 \ Xk+1 ) ∪ Xk+1 ) = µk (Yk+1 )).
Thus, we have constructed the family of probability πn -quasiinvariant measures (Pi )i∈N for which conditions 1) − 3) hold. On the one hand, it is easy to verify that the family of probability measures (Pi )i∈N is consistent in a wide sense. On the other hand, this family is not consistent in a narrow sense. Indeed, ∞ ∑ ∑ E= Xi and 1 = P0 (E) ̸= Pi (Xi ) = 0. i=1
i∈N
Example 2. An example of a consistent in a narrow sense family of probability measures which is not regular. Let E = [0; 1] × [0; 1], 1 1 1 1 S1 = {E, ∅, [ ; 1] × [ ; 1], E \ [ ; 1] × [ ; 1]}, 2 2 2 2 1 1 P1 ([ ; 1] × [ ; 1]) = 1, 2 2 1 1 1 1 S2 = {E, ∅, [0; [×[0; [, E \ [0; [×[0; [}, 2 2 2 2 1 1 P2 ([0; [×[0; [) = 1. 2 2 It is clear that each of the functionals P1 and P2 can be extended in a natural way onto the corresponding σ algebras S1 and S2 . We denote their extensions by the same symbols P1 and P2 . It is easy to verify that the family of probability measures (Pi )1≤i≤2 is consistent in a narrow sense but it is not regular. Indeed, if P is a measure defined on the σ-algebra σ(∪2i=1 Si ) such that 213
(∀i)(1 ≤ i ≤ 2 → P |Si = Pi ), then
1 1 1 1 P (([0; [×[0; [) ∪ ([ ; 1]) × [ ; 1])) = 2 2 2 2 1 1 1 1 = P ([0; [×[0; [) + P ([ ; 1[×[ ; 1[) = 2 2 2 2 1 1 1 1 = P2 ([0; [×[0; [) + P1 ([ ; 1[×[ ; 1[) = 1 + 1. 2 2 2 2 This is a contradiction, and the nonregularity of the family (Pi )1≤i≤2 is proved. Example 3. An example of a regular family of probability measures which is not extendable. Let E = [0; 1] and S = B([0; 1]). Denote by (Xi )i∈I the family of all first category Borel subsets of [0;1]. Let Si = {∅, E, Xi , E \ Xi }. Let us define the probability measure Pi on Si by the formula Pi (Xi ) = 0, Pi (E \ Xi ) = 1, Pi (∅) = 0, P (E) = 1. It is easy to verify that the family (Pi )i∈I is regular. Notice that the probability measure Pσ generated by the family of probability measures (Pi )i∈I has the form { 1, if [0, 1] \ Y is a set of first category, Pσ (Y ) = 0, if Y is a set of first category for arbitrary Y ∈ σ(∪i∈I Si ). Let us show that the probability measure Pσ is not extendable from the σ-algebra σ(∪i∈I Si ) on to the σ-algebra S. Indeed, if P is such an extension of Pσ , then, according to one result of Ulam (see Section 18), we easily conclude that the measure P is concentrated on the some first category Borel set X ⊂ [0; 1]. Hence we obtain a contradiction with the property of the measure Pσ , and the impossibility to extend Pσ onto σ-algebra S is proved. The following theorem contains a solution of the General Marginal Problem. Theorem 1. Let (E, S) be a measurable space and let (E, Si , Pi )i∈I be a family of probability spaces, such that (∀i)(i ∈ I → Si ⊂ S). 214
Then there exists a probability measure P defined on the measurable space (E, S) such that (∀i )(i ∈ I → P |Si = Pi ) if and only if the following two conditions hold: 1) the family of probability measures (Pi )i∈I is regular; 2) the probabality measure Pσ generated by ∪ the family of probabality measures (Pi )i∈I is extendable from the σ-algebra σ( Si ) on to the σ-algebra S. i∈I
Proof. Necessity. From the condition (∀i)(i ∈ I → Si ⊂ S), we have σ(∪i∈I Si ) ⊂ S. It is evident that Pσ = P |σ(∪i∈I Si ) is a probability measure such that P σ |S i = P i . It is also clear that P is an extension of the probability measure Pσ from the σ-algebra σ(∪i∈I Si ) to the σ-algebra S. Sufficiency. Let (Pi )i∈I be a family of probability measures for which conditions 1) − 2) hold. Denote by Pσ the probability measure generated by the family (P∪ i )i∈I . Let P be a measure extending the measure Pσ from the σ− algebra σ( Si ) to the σ−algebra S. Then we obtain: i∈I
A) dom(P ) = S, B) (∀i )(i ∈ I → P |Si = Pi ) and the sufficiency of Theorem 1 is proved. One sufficient condition for the regularity of a consistent in a narrow sense family of probability measures is given by Theorem 2. Let (Sξ )ξ≺η be a family of σ-algebras of subsets of the basic space E, such that (∀η1 )(∀η2 )(0 ≼ η1 ≺ η2 ≺ η → Sη1 ⊂ Sη2 ). Further, let (µξ )ξ≺η be a consistent in a narrow sense family of probability measures, such that 215
(∀ξ)(0 ≼ ξ ≺ η → dom(µξ ) = Sξ ). If an ordinal number ω is not cofinal with an ordinal number η, then ∪ S= Sξ ξ≺η
is a σ-algebra and the family of probability measures (µξ )ξ≺η is regular. Proof. Let us show that S = ∪ξ≺η Sξ is a σ−algebra. Let (Ak )k∈N be a countable family of elements of ∪ξ≺η Sξ . This means that (∀k)(k ∈ N → (∃ξk )(0 ≼ ξk ≺ η → Ak ∈ Sξk ). As the ordinal number ω is not cofinal with the ordinal number η, there exists an ordinal number ξ ≺ η such that (∀k)(k ∈ N → ξk ≺ ξ). It is clear that (∀k)(k ∈ N → Sξk ⊂ Sξ ). That is why ∪k∈N Ak ∈ Sξ ⊂ S, ∩k∈N Ak ∈ Sξ ⊂ S. Let us define the measure P by (∀A)(A ∈ S → P (A) = µξA (A)), where ξA denotes the first ordinal number for which A ∈ SξA . It is easy to verify that P is a probability measure which satisfies all conditions of Theorem 2. The following theorem is of some interest. Theorem 3. Let (Sξ )ξ≺η be a family of σ-algebras of subsets of a space E. Further, let (Pξ )ξ≺η be a consistent in a wide sense family of probability measures, such that (∀ξ)(ξ ≺ η → dom(Pξ ) = Sξ ). If the conditions 1) (∀ξ1 )(∀ξ2 )(0 ≼ ξ1 ≺ ξ2 ≺ ξ → Sξ1 ⊂ Sξ2 ), 2) an ordinal number ω is not cofinal with the ordinal number η, hold, then the family of probability measures (Pξ )ξ≺η is regular if and only if it is consistent in a narrow sense. 216
Proof. The necessity is obvious. Let us prove the sufficiency. Note that the functional P defined by (∀A)(A ∈ ∪ξ≺η Sξ → P (A) = PξA (A)), where ξA is the first ordinal number from the segment [0, η[ satisfying the condition A ∈ SξA , is a probability measure on the σ-algebra ∪ξ≺η Sξ . From the Carath´eodory theorem it readily follows that the measure P can be extended from the algebra ∪ξ≺η Sξ onto the σ-algebra σ(∪ξ≺η Sξ ) = ∪ξ≺η Sξ . Theorem 3 is proved. Theorem 4. Let (E, Sξ , Pξ )ξ≺η be a family of probability spaces. Further, let (Pξ )ξ≺η be a consistent in a narrow sense family of probability measures. If (∀(ξk )k∈N )((∀k )(k ∈ N → 0 ≺ ξk ≺ η) → (∃η0 )(η0 ≺ η & ∪k∈N Sξk ⊂ Sη0 )), then ∪ξ≺η Sξ is a σ-algebra, on which we can define a probability measure P such that (∀ξ)(0 ≺ ξ ≺ η → P |Sξ = Pξ ); Consequently, the family of probability measures (Pξ )ξ≺η is regular. Proof. Let us show that ∪ξ≺η Sξ is a σ−algebra. Indeed, if (∀k)(k ∈ N → Bk ∈ Sξk ), then the results of set-theoretical operations belong to Sη0 . This means that ∪ξ≺η Sξ
∪
,
∩
, \ applied to Bk (k ∈ N )
is the σ−algebra. Let us define a measure P by (∀A)(A ∈ ∪ξ≺η Sξ → P (A) = PξA (A)), 217
where ξA denotes the first ordinal number from the segment [0; η[ for which the condition A ∈ SξA holds. According to the consistency in a narrow sense of the family of probability measures (Pξ )ξ≺η , we conclude that P is a probability measure on ∪ξ≺η Sξ such that (∀ξ)(0 ≺ ξ ≺ η → P |Sξ = Pξ ). This completes the proof. Let us consider some examples. Example 4. Let E = [0; 1], S1 = {Y |Y ⊂ [0; 1] & Y ∈ B[0, 1] & (Y or [0; 1] \ Y is a first category set)}, S2 = B([0, 1]). Assume (∀Y )(Y ∈ S1 → { P (Y ) =
0, if Y is a set of first category, 1, if Y is a set of second category.
Clearly, the measure P is not extendable from the σ−algebra S1 onto the σ−algebra S2 . Example 5. Let E = [0; 1] × [0; 1], 1 1 S1 = {[0; [×[0; 1], [ ; 1] × [0; 1], E, ∅}, 2 2 1 1 P1 ([0; [×[0; 1]) = , 2 2 1 1 P1 (( ; 1] × [0; 1]) = , 2 2 1 1 S2 = {[0; 1] × [0; [, [0; 1] × [ ; 1], E, ∅}, 2 2 1 1 P2 ([0; 1] × [0; [) = , 2 2 218
1 1 P2 ([0; 1] × [ ; 1]) = . 2 2 It is easy to verify that the family of probability measures (Pi )1≤i≤2 is consistent in a wide sense. Let us denote the probability measure Pa by 1 1 1 1 1 Pa ([0; [×[0; [) = a, Pa ([0; [×[ ; 1]) = − a, 2 2 2 2 2 1 1 1 1 1 Pa ([ ; 1] × [0; [) = − a, Pa ([ ; 1] × [ ; 1]) = a 2 2 2 2 2 for all a ∈]0; 12 [. It is clear that the family (Pi )1≤i≤2 is regular, and each of the measures (Pa )a∈]0;1[ can be considered as a measure generated by the regular family (Pi )1≤i≤2 . Hence the measure generated by the above regular family is not unique. Theorem 5 (Necessary condition for the regularity). If a consistent in a narrow sense family of probability measures (Pi )i∈I is regular, then (∀(Xik )k∈N )((∀k)(k ∈ N → (ik ∈ I&Xik ∈ ∪i∈I dom(Pi ) & (Xik )k∈N is the family of disjoint sets →
∞ ∑
PXik (Xik ) ≤ 1))),
k=1
where PXik denotes a probability measure whose index is equal to a first ordinal number iXik for which the condition Xik ∈ SiXi
k
holds. Proof. Let a family (Pi )i∈I of probability measures be regular and P be the probability measure generated by this family. Then ∪k∈N Xik ∈ dom(P ). Hence we have ∞ ∑ k=1
PXik (Xik ) =
∞ ∑
P (Xik ) = P (
k=1
∞ ∑
k=1
Q. E. D. The following problem is interesting to solve. 219
Xik ) ≤ P (E) = 1.
Problem 2. Is the condition formulated in Theorem 5 sufficient for a consistent in a narrow sense family of probability measures to be regular? Let us consider the next theorem. Theorem 6. Let (E, Si , Pi )i∈I be a consistent in a narrow sense family of probability measures. Let S be an σ−algebra of subsets of the space E, such that (∀i)(i ∈ I → Si ⊂ S). Then there exists a unique probability measure defined on the σ−algebra S and satisfying the condition (∀i )(i ∈ I → P |Si = Pi ), if the probability measure P0 , defined by the formula ∩ (∀X)(X ∈ Si → P0 (X) = Pi (X)), i∈I
possesses a strong extension from the σ-algebra ∩i∈I Si onto the σ-algebra S. Proof. Let us denote by P an extension of the probability measure P0 from the σ-algebra ∩i∈I Si onto the σ-algebra S. Then (∀S)(∀P )((∩i∈I Si ⊂ S ⊂ S) & (P is an extension of the measure P0 from the σ − algebra ∩i∈I Si to the σ − algebra S) → P |S = P ). In particular, if S = Si (i ∈ I), we have P |Si = Pi . Let us show that the probability measure P is unique. Assume the contrary and let P1 be a probability measure on S such that: A) P1 ̸= P ; B) (∀i )(i ∈ I → P1 |Si = Pi ). The relation P1 ̸= P means that (∃X)(X ∈ S → P1 (X) ̸= P (X)). Denote by S ∗ the σ-algebra generated by the element X and the σ-algebra ∩i∈I Si . 220
Using the property of the measure P, from the condition ∩i∈I Si ⊂ S ∗ ⊂ S, we have P (X) = P1 (X), which contradicts the assumption P1 (X) ̸= P (X). Q. E. D. Remark 3. Definitions 1), 2), 3), 4), 5) can be generalized for an arbitrary family of σ-finite measures. Let us consider the following theorem. Theorem 7. Let (µi )i∈I be a consistent in a narrow sense family of σ-finite measures defined on the space E and (∀i )(i ∈ I → dom(µi ) = Si ). Further, let (∀i)(i ∈ I → Si is σ − ring of subsets of the space E). If the condition (∀(ik )k∈N )((∀k )( k ∈ N → ik ∈ I) → (∃i∗ )(i∗ ∈ I → Si∗ ⊂ ∪k∈N Sik )) holds, then there exists a unique measure µ on the σ-ring generated by ∪i∈I Si such that (∀i )(i ∈ I → µ|Si = µi ).
Proof. Note that ∪i∈I Si is a σ-ring of subsets of E. Indeed, let (Ak )k∈N be a family from ∪i∈I Si . This means that (∃(ik )k∈N ) (∀k) (k ∈ N → ik ∈ I & Ak ∈ Sik ). It is clear that there exists an index i∗ ∈ I such that ∪k∈N Sik ⊂ S ∪i∗ . Therefore, the results of an arbitrary countable quantity of operations ” ”, ∩ ” ”, ”\” over the elements of the family (Ak )k∈N also belong to the σ-ring Si∗ ⊃ ∪i∈I Si . Let us define the functional P by the formula (∀A)(A ∈ ∪i∈I Si → P (A) = PiA (A)), where iA = τi (i ∈ I & A ∈ Si ). 221
Further, if (Ak )k∈N is a family of disjoint subsets, then by the consistency in a narrow sense of the family (Pi )i∈I , we have P(
∞ ∑ k=1
Ak ) = Pi∗ (
∞ ∑
Ak ) =
k=1
∞ ∑
Pi∗ (Ak ) =
k=1
∞ ∑
P (Ak ).
k=1
An application of the Carath´eodory theorem completes the proof of Theorem 7. Q. E. D. Remark 4. Applying Theorem 7, we conclude that the well-known Kolmogorov theorem (see, e.g. [22],[36],[123]) is a statement that an arbitrary family of probability measures generated by the consistent family of all standard countable-dimensional distributions defined in Rα is consistent in a narrow sense, for an arbitrary nonepty parametric set α. Let us consider one application of the above-mentioned results in the theory of invariant measures. Theorem 8. For an arbitrary parameter set α, in the topological vector space Rα there exists an R(α) -invariant measure µ which takes value 1 on the set [0, 1]α such that dom(µ) ⊂ B(Rα ). Proof. Let J ⊂ α be an arbitrary countable subset of the parametric set α. (1) Define by λα\J the probability Haar measure defined on the set [0, 1]α\J . Let (2)
us denote by λJ the measure µ[0,1]J (see Section 6). Suppose that (1)
(2)
(∀J)(J ⊂ α & card(J) ≤ ℵ0 , → (µJ = λα/J × λJ ) & (SJ = dom(µJ ))). Since for the family of measures (µJ )
J⊂α
card(J)≤ℵ0
all conditions of Theorem 7 hold, on the σ-ring ∪J⊂α,card(J)≤ℵ0 SJ there exists a unique measure µ such that (∀J)(J ⊂ α & card(J) ≤ ℵ0 → µ|Sη = µη ). Let us show that the measure µ is R(α) -invariant. Indeed, g ∈ Rα ⇐⇒ (∃n)(∃(gik )1≤k≤n )(n ∈ N & ((∀k)(k ∈ N → ik ∈ α & { 0, if i ̸∈ ∪nk=1 ik , gik ∈ R) → (∀i )(i ∈ α → Pri (g) = gik , if i = ik . 222
Let us consider an arbitrary set A ∈ ∪J⊂α,card(J)≤ℵ0 SJ . Then there exists a countable subset J0 ⊂ α such that A ∈ SJ0 . Let us put J1 = J0 ∪ {∪nk=1 ik }. On the one hand, SJ1 ⊂ SJ0 . On the other hand, the measure µJ1 is an R(J1 ) -invariant σ-finite measure. That is why µ(g(A)) = µJ1 (g(A)) = µJ1 (A) = µ(A). The relation dom(µ) ⊂ B(Rα ) is clear. Q. E. D. Note that the measure µ for card(α) > ℵ0 is not σ-finite. Moreover, it has not got the Suslin property and the maximal cardinality of a family of disjoint subsets with positive µ-measure is equal to α. Remark 5. Using Theorem 1 from Section 6, we easily conclude that the vector space of all admissible translations G in the sense of invariance of the measure µ has the form G = {g : α → R | (∃(ik )k∈N )(∃gik )(ik ∈ α & gik ∈ R → { (∀i)(i ∈ I → g(i) =
& (∃k0 )(k0 ∈ N → the series
gik , 0, ∑
if i = ik , if i ∈ / α \ ∪k∈N {ik } ln(1 − |g(ik )|) is convergent ))}.
k≥k0
The following unsolved problem is of interest from measure-theoretical point of view. Problem 3. Let card(α) > ℵ0 . Does there exist an R(α) -invariant extension of the measure µ from the σ-ring ∪J⊂α,card(J)≤ℵ0 SJ onto the σ-algebra B(Rα )?
223
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