International Journal of Pure and Applied Mathematics

International Journal of Pure and Applied Mathematics ————————————————————————– Volume 24 No. 3 2005, 299-322

HYPERMEASURES IN GENERAL SPACES M. Burgin Department of Mathematics University of California 405 Hillgard Avenue, P.O. Box 951555 Los Angeles, CA 90095-1555, USA

Abstract: The theory of hypernumbers is a novel approach in functional analysis aimed at the development of such mathematically correct technique, which allows operations with divergent integrals and series. Although, it resembles nonstandard analysis, there are several distinctions between these theories. For example, while nonstandard analysis changes spaces of real and complex numbers by injecting in to them infinitely small numbers and other nonstandard entities, the theory of extrafunctions does not change the inner structure of spaces of real and complex numbers, but adds to them infinitely big numbers as external objects. The goal of this paper is to develop the theory of hypermeasures in a sufficiently general context. To extend the scope of integration, we construct various kinds of hypermeasures. Hypermeasures similar to measures are set functions, only in contrast to measures, their values are not only numbers, but also arbitrary hypernumbers. AMS Subject Classification: 28B99 Key Words: measure, hypernumber, hypermeasure, set ring, measurable set, linear space

1. Introduction Measures play an important role in calculus and modern functional analysis, forming the base for the theory of integration. However, the classical concept of a measure (as well as its modern generalizations) has definite limitations. For Received:

June 5, 2005

c 2005, Academic Publications Ltd.

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instance, having a natural measure µ in a real line R such as Lebesgue measure, we cannot naturally define the measure of the whole R or the infinite interval [0, ∞]. By monotonicity of measure, these measures cannot be real numbers. If we define, as it is often done, µ (R) = ∞ and µ ([0, ∞]) = ∞, then the difference between the whole line and its half disappears. Limitations of the concept of a measure result in restrictions on applications of integral calculus. At the same time, some important problems of contemporary physics are related to measures. For instance, Feynman path integrals provide a powerful tool for different domains of modern physics [11, 14, 21]. However, the mathematical theory underlying lots of physical calculations with path integrals is far from being complete (cf., for example, [13, 18]). It is possible to divide all approaches to a rigorous mathematical construction of the path integral into two classes [17]. In the approaches from the first class, there is no measure to define and evaluate path integrals by a conventional technique that exists in calculus. Path integral is considered as a generalized functional on an appropriate space of functions. These approaches cover only a very restricted class of path integrals. In the approaches from the second class, path integrals are defined as a conventional integral over some measure on an appropriate space of trajectories. However, these methods also work under very restrictive conditions. Utilization of hypernumbers [5]-[7] allows one to extend the concept of measure to the concept of hypermeasure, making it possible to measure much broader scope of sets than by ordinary measures, as well as to extend the conventional construction of integration, including functional and, in particular, path integration. This extension is constructed in a natural way. Informally, hypermeasures are “measures” that take values in hypernumbers. As a result, all measures are finite hypermeasures and many properties of hypermeasures are similar to properties of measures. For instance, we can extend a positive hypermeasure from a set semiring to a set ring (Theorem 3.1), hypermeasures form a linear space (Proposition 4.1) and positive hypermeasures form a convex cone in this space (Proposition 4.2). We also build Jordan decomposition (Theorem 4.3) and Hahn decomposition (Theorem 4.4) of hypermeasures. As a result, it becomes unnecessary to prove many known properties of measures because they become direct corollaries of the corresponding results for hypermeasures. At the same time, there are properties that distinguish hypermeasures from measures. For instance, there are no σ-additive hypermeasures that are different from σ-additive measures. In addition to hypermeasures and measures, we also consider here fuzzy hypermeasures and measures. To make this paper selfcontained, in Section 2, going after Introduction, elements of the theory of

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hypernumbers are presented. Denotations. N is the set of all natural numbers; ω is the sequence of all natural numbers; ∅ is the empty set; R is the set of all real numbers; Q is the set of all rational numbers; R + is the set of all non-negative real numbers; R ++ is the set of all positive real numbers; R ω is the set of all sequences of real numbers; if a is a real number, then |a| or ||a|| denotes its absolute value or modulus; if a is a real number and t ∈ R ++ , then O t a = { x ∈ R; a – t < x < a + t } is a neighborhood of a; ρ(x, y) = |x − y| for x, y ∈ R; if X is a set, then 2 X is the set of all subsets of X. If X and Z are sets, then X∆Z = X\Z ∪ Z\X; if a = (ai )i∈ω is a sequence of real numbers, then α = Hn(ai )i∈ω is the real hypernumber determined by a; R ω is the set of all real hypernumbers; R + ω is the set of all non-negative real hypernumbers, i.e., hypernumbers that are larger than or equal to zero; R ++ is the set of all real ω hypernumbers that are positive, i.e., larger than zero.

2. Elements of the Theory of Hypernumbers Let R ω = { (ai )i∈ω ; ai ∈ R} be the set of all sequences of real numbers. Definition 2.1. For arbitrary sequences a = (ai )i∈ω , b = (bi )i∈ω ∈ Rω : a ∼ b means that lim

i→∞ |ai

− bi | = 0 .

Definition 2.2. Classes of the equivalence ∼ are called real hypernumbers and their set is denoted by R ω . Any sequence a = (ai )i∈ω determines a hypernumber α = Hn(ai )i∈ω . It is possible to build in the same way complex hypernumbers. They are sets of equivalent sequences of complex numbers. Real numbers contain different subclasses: rational, irrational, transcedental, integer, etc. In the universe of hypernumbers Rω , there are even more subclasses. For instance, it is possible to introduce three types of hypernumbers: stable, infinite, and oscillating hypernumbers. Example 2.1. An infinite increasing hypernumber: α = Hn(ai )i∈ω , where ai = i, i = 1, 2, . . . . Example 2.2. An infinite increasing hypernumber: β = Hn(bi )i∈ω , where bi = 2 i , i = 1, 2, . . . . Example 2.3. A finite (bounded) oscillating hypernumber: γ = Hn(ai )i∈ω , where ai = (−1)i , i = 1, 2, . . . . Example 2.4. An infinite oscillating hypernumber: δ = Hn(ai )i∈ω , where ai = (−1)i · i, i = 1, 2, . . . .

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Example 2.5. An infinite decreasing hypernumber: ν = Hn(ai )i∈ω , where ai = -5 i, i = 1, 2, . . . Example 2.6. An infinite oscillating hypernumber: θ = Hn(ai )i∈ω , where ai =2 i + (−1)i · i, i =1, 2, . . . . Here we give exact definitions for classes of hypernumbers. Definition 2.3. A real hypernumber α is called finite or bounded if there is a sequence (ai )i∈ω such that α = Hn(ai )i∈ω and for some positive real number b, we have |bi | < b for almost all i ∈ ω. Lemma 2.1. The following conditions are equivalent: a) α is a finite real hypernumber; b) there is a sequence (ai )i∈ω such that α = Hn(ai )i∈ω and for some real number b, |bi | < b for all i ∈ ω; c) for any sequence (ai )i∈ω such that α = Hn(ai )i∈ω there is a real number b such that | bi | < b for almost all i ∈ ω; d) for any sequence (ai )i∈ω such that α = Hn(ai )i∈ω there is a real number b such that | bi | < b for all i ∈ ω. / Definition 2.4. A real hypernumber α = Hn(ai )i∈ω is called proper if α ∈ R. Improper real hypernumbers are exactly real numbers. Definition 2.5. A real hypernumber α = Hn(ai )i∈ω is called stable if there are a real number b and a sequence (ai )i∈ω such that α = Hn(ai )i∈ω and bi = b for almost all i ∈ ω. For such a hypernumber, we have α = b. Lemma 2.2. Any stable hypernumber α = Hn(ai )i∈ω is finite. Lemma 2.3. The following conditions are equivalent: a) α is a stable real hypernumber; b) there is a sequence (ai )i∈ω such that α = Hn(ai )i∈ω and for some real number b, bi = b for all i ∈ ω; c) for any sequence (ai )i∈ω such that α = Hn(ai )i∈ω , there is a real number b such that bi = b for almost all i ∈ ω; d) for any sequence (ai )i∈ω such that α = Hn(ai )i∈ω , there is a real number b such that bi = b for all i ∈ ω. As stable real hypernumbers that correspond to different real numbers b cannot be equal, we have the following result. Lemma 2.4. There is a one-to-one correspondence fR between R and the subset StR ω of all stable hypernumbers from R ω , which is defined by the formula fR (a) = = Hn(ai )i∈ω with all ai = a. In what follows, we will identify stable real hypernumbers and corresponding real numbers. For example, (8, 8, 8, . . . , 8, . . . ) = 8 in R ω .


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Definition 2.6. A real hypernumber α = Hn(ai )i∈ω is called oscillating if there is such k ∈ R ++ that there are two infinite sequences of natural numbers m(i) and n(i) with i = 1, 2, . . . such that am(i) – an(i) > k and an(i) – am(i+1) > k for all i = 1, 2, . . . Remark 2.1. Oscillating hypernumbers may be bounded or finite (cf. Example 2.3) and unbounded or infinite (cf. Example 2.4 and Example 2.6). Definition 2.7. A real hypernumber α = Hn(ai )i∈ω is called: 1) infinite increasing if ∃ j ∈ ω∀i¿j (ai+1 – ai > 0)) & (∀p ∈ R ∃ i ∈ ω (ai > p)); 2) infinite decreasing if ∃ j ∈ ω∀i¿j (ai+1 – ai > 0)) & (∀p ∈ R ∃ i ∈ ω (ai > p)); 3) infinite expanding if there are subsequences (bi )i∈ω and(ci )i∈ω of the sequence (ai )i∈ω such that Hn(bi )i∈ω is an infinite increasing and Hn(ci )i∈ω is an infinite decreasing hypernumber; 4) strictly infinite expanding if it is infinite expanding and (ai )i∈ω = (bi )i∈ω ∪ (ci )i∈ω ; 5) infinite monotonous if α = Hn(bi )i∈ω for some monotonous sequence { bi ; i = 1, 2, . . . } Proposition 2.1. Any finite real hypernumber is either a real number or an oscillating real hypernumber. Proposition 2.2. Any infinite real hypernumber is either an infinite increasing hypernumber or an infinite decreasing hypernumber or an oscillating real hypernumber. Remark 2.2. For complex hypernumbers in general, Proposition 2.2 and Proposition 2.3 are not true. The following lemma gives characteristics properties of real hypernumbers that are outside the real line. Lemma 2.5. A real hypernumber α = Hn(ai )i∈ω is proper, that is, α ∈ / R, if and only if either the sequence a = (ai )i∈ω is unbounded or (condition D) there is an interval (d, b) such that there are infinitely many elements larger than b and infinitely many elements smaller than d. Proof. 1. Let α ∈ R. Then by Lemma 2.3, lim i→∞ ai = a ∈ R. Consequently [22], the sequence a = (ai )i∈ω is bounded. In addition, for any pair of real numbers (d, b), we have either a ≤ d or d < a < b or b ≤ a. In the first case, almost all elements ai are smaller than b as d is strictly smaller than b. In the second case, almost all elements ai are larger than d and smaller than b. In the third case, almost all elements ai are larger than d as d is strictly smaller than b. So, the condition of the lemma is not satisfied. 2. For any real hypernumber α = Hn(ai )i∈ω , we have two cases: the se-

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quence a = (ai )i∈ω is bounded or unbounded. So, to prove the lemma, we need to consider only the case when this sequence is bounded. It means that all elements ai belong to some interval [u, v]. Let us divide [u, v] into three equal parts: [u, u1 ], [u1 , u2 ], and [u2 , v]. Then either almost all elements ai belong to only one of these intervals (say, [u, u1 ]) or to two adjacent intervals (say, [u, u1 ] and [u1 , u2 ]) or to two non-adjacent intervals to all three intervals. In the two latter cases, everything is proved because we can take (u1 , u2 ) as (d, b). Otherwise we continue decomposition of intervals: in the first case, of the interval [u, u1 ] and in the second case, of the interval [u, u2 ]. We continue this process. If at some step, we get the case three or four, then we have an interval (d, b) that is necessary for validity of the lemma. If we always get cases one or two, then it gives us a system of imbedded closed intervals, the length of which converges to 0 and each of them contains almost all elements ai . By the standard argument, this implies that the sequence a = (ai )i∈ω has the limit. Consequently, α ∈ R. Thus, if α ∈ / R, the condition of the lemma has to be true. Lemma 2.5 is proved. Lemma 2.5 allows one to give a complete characterization of hypernumbers. Theorem 2.1. There are four disjoint classes of hypernumbers and each hypernumber belongs to one of them: — Stable hypernumbers. — Bounded oscillating hypernumbers. — Unbounded oscillating hypernumbers. — Infinite monotonous hypernumbers. In turn, unbounded oscillating hypernumbers form three groups: — Bounded from above oscillating hypernumbers. — Bounded from below oscillating hypernumbers. — Two-way unbounded oscillating hypernumbers. Infinite hypernumbers form two groups: — Positive infinite monotonous hypernumbers. — Negative infinite monotonous hypernumbers. Definition 2.8. A sequence (ai )i∈ω is a subsequence of a sequence (bi )i∈ω if there is a strictly increasing function g: N → N such that for any i ∈ N , we have ai = bg(i) . A more general nature of real hypernumbers, in comparison with real numbers, brings us to new constructions related to hypernumbers. Definition 2.9. A real hypernumber α = Hn(ai )i∈ω is called a subhypernumber of a real hypernumber β = Hn(bi )i∈ω if the sequence (ai )i∈ω is a


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subsequence of the sequence (bi )i∈ω . We denote this by αβ. Sub α = { γ; γα } denotes the set of all subhypernumbers of a real hypernumber α. Definition 2.10. If α 6= β, then α is called a proper subhypernumber of β (α ≺ β). The concept of a subhypernumber does depend on the choice of a representing sequence as it is demonstrated by the following result. Lemma 2.6. If β = Hn(bi )i∈ω = Hn(ci )i∈ω and α = Hn(ai )i∈ω , where (ai )i∈ω is a subsequence of (bi )i∈ω , then there is a subsequence (di )i∈ω of (ci )i∈ω such that α = Hn(di )i∈ω . Proposition 2.3. α = Hn(ai )i∈ω is an oscillating real hypernumber if and only if it has an oscillating proper subhypernumber. Relations on R induce corresponding relations on R ω . Definition 2.11. For any a, b ∈ R ω , a ≤ b if ∃ n ∀ i ≥ n (ai ≤ bi ) and a < b if ∃ n ∀ i ≥ n (ai < bi ). These relations induce similar relations on R ω : Definition 2.12. For any α, β ∈ R ω , α ≤ β if ∃a ∈ α∃b ∈ β (a ≤ b) and α < β if (∃a ∈ α∃b ∈ β (a < b)) & α 6= β. It is proved (cf., for example, [7]) that relations ≤ and < on R ω and R ω are a partial order and a strict partial order, respectively. If X is a partially ordered set with the order relation ¡, then it is possible to define a partial order for subsets of X. Definition 2.13. For any A, B ⊆ X, A < B if ∀a ∈ A∀b ∈ B (a < b). In such a way, we obtain a partial order called the induced order relation on sets of hypernumbers. Definition 2.14. The component C α of a hypernumber α is the set of all hypernumbers β such that the difference α − β is finite. For instance, the component of zero C0 is the set of all finite real hypernumbers, which contains the set R of all real numbers. Lemma 2.7. For any β ∈ C α, C α = C β. For any subspace X, of the space R ω , a component of X is the intersection of X with some component C α. Lemma 2.8. Any two components either coincide or do not intersect. Indeed, if α 6= β and C α∩ C β 6= ∅, then there a hypernumber γ such that γ ∈ C α∩ C β. Consequently, α − γ and β − γ are finite hypernumbers. Then α − β = (α − γ) - (β − γ) is also a finite hypernumber. Thus, by definition β ∈ C α, and by Lemma 2.8, C α = C β. Proposition 2.4. For any hypernumber α, C α = C α + R. Definition 2.14. The strict component SC α of a hypernumber α is the set of all hypernumbers β such that the difference α - β is a real number.

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For instance, the strict component of zero SC0 is the set R of all real numbers. Lemma 2.9. For any β ∈ SC α, SC α = SC β. For any subspace X, of the space R ω , a strict component of X is the intersection of X with some strict component SC α. Lemma 2.10. Any two strict components either coincide or do not intersect. Proposition 2.5. For any hypernumber α, SC α = SC α + R. Proposition 2.6. For any hypernumber α, its strict component SC α is isomorphic to R as a partially ordered set. As R is linearly ordered, we have the following result. Corollary 2.1. Any strict component SC α is linearly ordered.

3. Positive Hypermeasures To build a definite integral with values in hypernumbers called hyperintegral, we develop a theory of hypermeasures. At first, we consider positive hypermeasures and fuzzy hypermeasures. Let X be a set and B ⊆ 2 X . + ) is called a Definition 3.1. A function µ: B → R + ω (µ: B → R [linearly] positive fuzzy hypermeasure (fuzzy measure) in X if [µ is a mapping in some linearly ordered subset of R ω and] the following conditions are satisfied: (F1) ∅ ∈ B. (F2) A, B ∈ B implies A ∩ B ∈ B. (F3) If D ⊆ A and A, D ∈ B, then A = D∪ (∪kn=1 An ) for some k ≥ 0 where D ∩ Ai = ∅, Ai ∩ Aj = ∅ for all i, j ≤ k, and A1 , . . . , Ak ∈ B. (F4) For any A and B from B, the inclusion A ⊆ B implies µ(A) ≤ µ(B), i.e., µ is monotone. Remark 3.1. According to Kolmogorov and Fomin [16], systems of sets that satisfy conditions (F1) – (F3) are called set semirings. Example 3.1. The set CI of all closed intervals [a, b] in the real line R is a set semiring. Example 3.2. The set OI of all open intervals (a, b) in the real line R is a set semiring. Example 3.3. We define the function µ: CI → R + by the following formula: µ ([a, b]) = max { |n|; n ∈ [a, b] ∩ N }.


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µ is a positive fuzzy measure in R. However, it is not a positive measure (cf. Definition 3.10) as it is not additive. Example 3.4. The set IS of all finite and infinite open intervals (a, b) in the extended real line R ∪ { ∞ } is a set semiring. Example 3.5. We define the function µ: IS → R + by the following formulas: If a, b ∈ R, then µ ((a, b)) = max { |n|; n ∈ [a, b] ∩ N }; µ ((0, ∞)) = Hn(i)i∈ω = µ ((a, ∞)) for all a < 0; If a ∈ R

+

and b = ∞, then µ ((a, ∞)) = µ ((0, ∞)) - µ ((0, a));

If a = - ∞ and b ∈ R

+,

then µ ((- ∞, b)) = µ ((0, b));

If b < 0, then µ ((a, b)) = 0. Thus, µ is a positive fuzzy hypermeasure in R. However, it is not a positive hypermeasure as it is not additive. Definition 3.2. Sets from B are called µ -hypermeasurable or simply fuzzy hypermeasurable. Definition 3.3. A set A from B with µ(A) ∈ R is called µ -measurable or simply fuzzy measurable. The set of all µ -measurable sets from B is denoted by B R . Let µ be a positive fuzzy hypermeasure in X. Lemma 3.1. If B is a set semiring and B R 6= ∅, then B R is also a set semiring. Proof. Condition (F1), i.e., ∅ ∈ B R , is true because ∅ ⊆ A implies µ(∅) ≤ µ(A) for all sets A, including those of them that belong to B R . Properties of B R entail Condition (F2): A, B ∈ B R implies A ∩ B ∈ B R because B is a set semiring and µ is monotone. Condition (F3): Let A, D ∈ B R . Then A = D∪ (∪kn=1 An ) for some k ≥ 0, where D ∩ Ai = ∅, Ai ∩ Aj = ∅ for all i, j ≤ k, and A1 , . . . , Ak ∈ B because A, D ∈ B. All A1 , . . . , Ak belong to B R because A1 , . . . , Ak ⊆ A and µ is monotone. Corollary 3.1. The restriction µR of µ to B R is a positive fuzzy measure in X. Let µ be a linearly positive fuzzy hypermeasure in X and the range R(µ) contains R. Lemma 3.2. If B ∈ B R , then all µ -hypermeasurable subsets of B are µ -measurable.

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Indeed, if D is a µ -hypermeasurable subset of B, then by monotonicity of µ, the value µ (D) is finite and this value cannot be an oscillating hypernumber because if we add an oscillating hypernumber to R, then the new set will not be linearly ordered, while by the initial conditions R(µ) is a linearly ordered set. Let us put B BR = 2 B ∩ B R , where B ∈ B R . Corollary 3.2. If B is a set semiring and B R 6= ∅, then B BR is also a set semiring for any B ∈ B R . Definition 3.4 [16]. A system of sets is called a set ring if it satisfies conditions (F2) and (F5), where: (F5) A, B ∈ B implies A∆B ∈ B, where A∆B = (A\ B)∪ (A\B). Lemma 3.3. (see [16]) For any set ring B, we have ∅ ∈ B and A, B ∈ B implies A ∪ B, A\B ∈ B. Definition 3.5 [16]. A set ring B with a unit element, i.e., an element E from B such that for any A from B, we have A ∩ E = A, is called a set algebra. Remark 3.2. In the case when B is an algebra of sets, X ∈ B, µ maps B into R + , and µ(X) = 1, Definition 3.1 specifies the construction of a fuzzy measure given in [15]. Popular examples of positive fuzzy measures and thus, of positive fuzzy hypermeasures are possibility, belief and plausibility measures. Example 3.6. A possibility measure in X is a positive fuzzy hypermeasure. Let us consider some set X and its power set P(X). Definition 3.6. A possibility measure in X is a partial function Pos: P(X) → [0,1] that is defined on a subset A from P(X) and satisfies the following axioms (cf. [18, 32, 33]): (Po1) ∅, X ∈ A, Pos(∅) = 0, and Pos(X) = 1. (Po2) For any A and B from A, the inclusion A ⊆ B implies Pos(A) ≤ Pos(B). (Po3) For any system { Ai ; i ∈ I } of sets from A, Pos(∪i∈I Ai ) = sup Pos(Ai )) .

(1)

i∈I

Example 3.7. Possibility is also described by a more general class of positive fuzzy measures (cf. [19, 25]). Definition 3.7. A quantitative possibility measure in X is a function P : P(X) → [0,1] that satisfies the following axioms: (Po1) ∅, X ∈ A, P (∅) = 0, and P (X) = 1. (Po5) For any A and B from P(X), P (A ∪ B) = max { P (A), P (B) }. Example 3.8. Dual to a quantitative possibility measure is a necessity measure [8, 23].


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Definition 3.8. A quantitative necessity measure in X is a function N : P(X) → [0,1] that satisfies the following axioms: (Ne1) N (∅) = 0, and N (X) = 1. (Ne2) For any A and B from P(X), N (A ∩ B) = min { N (A), N (B) }. Possibility and necessity measures are used in support logic programming, which uses fuzzy measures for reasoning under uncertainty and approximate reasoning in expert systems, based on logic programming style. One more example of fuzzy measures is given by quasi-measures [1], [2]. Definition 3.9. A function µ: B → R + is called a quasi-measure in X if the following conditions are satisfied: (F1) ∅ ∈ B. (F2a) A, B ∈ B implies A ∩ B ∈ B and A ∪ B ∈ B. (F4) For any A and B from B, the inclusion A ⊆ B implies µ(A) ≤ µ(B), i.e., µ is monotone. (A5) µ is additive, i.e., for any A and B from B, the equality A ∩ B = ∅ implies µ (A ∪ B) = µ(A) + µ(B) Remark 3.3. Initially quasi-measures are defined for the case when X is a compact Hausdorff space and B is the set of all open and closed subsets of X. Definition 3.10. A fuzzy (positive) hypermeasure µ in X is called: a) additive if for any A and B from B, the equality A ∩ B = ∅ implies µ(A ∪ B) = µ(A) + µ(B) .

(2)

b) super-additive if for any A and B from B, the equality A ∩ B = ∅ implies µ(A ∪ B) ≥ µ(A) + µ(B) .

(3)

c) subadditive if for any A and B from B, the equality A ∩ B = ∅ implies µ(A ∪ B) ≤ µ(A) + µ(B) .

(4)

d) σ− additive if for any A = ∪∞ n=1 An from B with Ai ∩ Aj = ∅ and An ∈ B for all i, j, n = 1, . . . , m, . . . with i 6= j, we have µ(A) = Σ∞ n=1 µ(An ) .

(5)

e) σ− subadditive if for any A ⊆ ∪∞ n=1 An from B and An ∈ B for all i, j, n =1, . . . , m, . . . we have µ(A) ≤ Σ∞ (6) n=1 µ(An ) . f) restrictedly σ -additive if for any A = ∪∞ n=1 An from B R with Ai ∩ Aj = ∅ and An ∈ B R for all i, j, n = 1, . . . , m, . . . with i 6= j, we have µ(A) = Σ∞ n=1 µ(An )

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g) restrictedly σ -subadditive if for any A ⊆ ∪∞ n=1 An from B for all i, j, n =1, . . . , m, . . . we have µ(A) ≤ Σ∞ n=1 µ(An )

R

and An ∈ B (8)

Definition 3.11. An additive fuzzy [linearly] positive hypermeasure µ in X is called a [linearly] positive hypermeasure in X, the pair (X, B) is called a measurable space, or hypermeasurable space, and the triad (X, B, µ) is called a hypermeasure space. If µ is a measure in X, then (X, B, µ) is called a measure space. Example 3.9. We define the function µ: IS → R + by the following formulas: If a, b ∈ R, then µ ([a, b]) = b − a; For any X ⊆ R, we put µ(X) = Hn(ai )i∈ω , where λ is the Lebesgue measure in R and an = λ ((X∩ [- n, n]) for all n = 1, 2, 3, . . . if the set X∩ [n, n] is λ -measurable. Properties of the Lebesgue measure and hypernumbers show that µ is a positive hypermeasure in R. It is a natural extension of the Lebesgue measure. Remark 3.4. All positive measures are linearly positive. Lemma 3.4. The sum of two [linearly] positive (fuzzy, super-additive, subadditive, σ-additive, σ -subadditive) hypermeasures is also a [linearly] positive (fuzzy, super-additive, subadditive, σ-additive, σ -subadditive, respectively) hypermeasure. Lemma 3.5. If µ is a [linearly] positive (fuzzy, super-additive, subadditive, σ-additive, σ -subadditive) hypermeasure in X and a is a positive real number, then aµ is also a [linearly] positive (fuzzy, super-additive, subadditive, σ-additive, σ -subadditive, respectively) hypermeasure in X. Lemma 3.6. Additivity of a fuzzy positive hypermeasure implies its superadditivity and super-additivity implies condition (F4) from Definition 3.1. Lemma 3.7. Any positive hypermeasure µ in X is monotone and thus, a positive fuzzy hypermeasure. Lemma 3.8. σ-additivity of a fuzzy positive hypermeasure implies its σsubadditivity. Let µ be a positive hypermeasure in X. Lemma 3.9. µ(∅) = 0. Indeed, ∅ ∪ ∅ = ∅. Thus, µ(∅) = 2 µ(∅). This is possible only if µ(∅) = 0. Lemma 3.10. If B is a set ring, then B R is also a set ring. Corollary 3.3. The restriction µR of µ to B R is a positive measure in X. Corollary 3.4. If B is a set ring (σ-algebra), then B BR is also a set ring (σ-algebra).


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Definition 3.12. Sets from B are called µ -hypermeasurable or simply hypermeasurable. Definition 3.13. A set A from B with µ(A) ∈ R is called µ -measurable or simply measurable. Remark 3.5. In the case when µ maps B into R + , Definition 3.11 gives us the construction of a measure from [16]. Remark 3.6. In some sources, a measure in X is called a premeasure, while a measure has to satisfy an additional condition of σ-additivity (cf., for example, [23] or [9]). Remark 3.7. In some sources, additive measures are called finitely additive, while σ-additive measures are called countably additive (cf., for example, [3]). σ-additive measures are more powerful than those measures that are only additive for developing probability theory. However, as Ash [3] writes, a convincing physical justification of σ-additivity has yet to be given. If the probability p(E) of the event E is to represent the long run relative frequency of E in a sequence of performances of a random experiment, p must be a finitely additive function. At the same time, only finitely many measurements can be made in a finite time interval, so σ-additivity is not inevitable on physical grounds. That is why here we assume that hypermeasures and measures are only additive and consider σ-additivity as an additional property of hypermeasures and measures. It is possible (cf., for example, [16]) to embed any set semiring Q into a set ring R(Q) such that it is a minimal set ring that contains Q. Theorem 3.1. a) For any positive hypermeasure µ in X defined on a set semiring B, there is a unique extension to a hypermeasure ν defined on the set ring R(B). b) If µ is σ-additive, then ν is also σ-additive. The proof is similar to the proof of the corresponding property for measures (cf. ([16], pp. 305 and 308)). Definition 3.14. (see [16]) A set algebra B closed with respect to countable unions of its elements is called a set σ -algebra or simply, σ -algebra. Remark 3.7. σ-algebras are considered in [23] under the name additive classes of sets. Two infinite operations (limits) are considered for sequences of sets. Let Q = { Ai ; i = 1, 2, 3, . . . } be a sequence of sets. Then ∞ lim inf Ai = ∪∞ k=1 ∩i=k Ai

and ∞ lim sup Ai = ∩∞ k=1 ∪i=k Ai .

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Thus, the lower limit lim inf Ai of the sequence Q is the set of all elements a belonging to all sets Ai from some i onwards. The upper limit lim sup Ai of the sequence Q is the set of all elements a such that a ∈ Ai holds for infinitely many sets Ai . Proposition 3.1. (see [23]) A σ-algebra B is closed with respect to lower and upper limits. Let (X, B, µ) be a hypermeasure space with the σ-additive positive hypermeasure µ. Proposition 3.2. If B R is a σ-algebra, then there is a number c ∈ R+ such that for any µ -measurable set A, we have µ(A) < c. Proof. Let us assume that for any n ∈ N , there is a µ -measurable set An with µ(An ) > n. Then taking D0 = ∅, we can build the sequence Dn = An \ An−1 ; i = 1, 2, 3, . . . of non-intersecting sets. In this sequence, all Dn belong to B and Dn ∩Dm = ∅ for any m, n = 1, 2, 3, . . . with n 6= m. Then the union D = ∪∞ n=1 Dn belongs to B R as B R is a σ-algebra. Thus, µ(D) ∈ R. In addition, µ(D) = Σ∞ n=1 µ(Dn ) because µ is a σ-additive positive hypermeasure. By the construction, Σ∞ n=1 µ(Dn ) > µ(Am ) > m for any m ∈ N . This contradiction concludes the proof. Corollary 3.5. Any σ-additive positive measure µ is bounded. Remark 3.8. When a measure is defined as a σ-additive function µ: B → R + ∪ { ∞ }, then µ can be unbounded.

4. General Hypermeasures Although the most popular measures are positive, in some cases, it is assumed that a measure maps a set rig or algebra B into R (cf., for example, [9]). This implies existence of a broader system of measures and suggests that it is reasonable to consider hypermeasures that are not necessarily positive. Definition 4.1. A function µ: B → R ω is called a hypermeasure in X and the triad (X, B, µ) is called a hypermeasure space if the following conditions are satisfied: a) B is a set semiring; b) if for any A, B, and A ∪ B from B, the equality A ∩ B = ∅ implies µ(A ∪ B) = µ(A) + µ(B) .

(9)

In this case, the pair (X, B) is called a measurable space, or hypermeasurable space, and the triad (X, B, µ) is called a hypermeasure space. If µ is a measure in X then (X, B, µ) is called a measure space.


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Remark 4.1. Bourbaki [4] also developed an approach, where measures are not only positive ones. Remark 4.2. In a general case, hypermeasures are not monotonous and Lemma 3.2 is not valid. Remark 4.3. σ-additive general measures are considered by Saks [23] under the name an additive function of sets, by Kolmogorov and Fomin [16] ˇ under the name an alternate measure or charge, and by Silov and Gurevich [24] under the name an alternate measure. Remark 4.4. In [10], such set functions are called charges. Remark 4.5. Parallel to quasi-measures, signed quasi-measures are defined as mappings µ: B → R that satisfy some additional conditions (cf., for example, [12]) for the case when X is a compact Hausdorff space and B is the set of all open and closed subsets of X. Lemma 4.1. If µ is a hypermeasure in X, then µ(∅) = 0. Indeed, ∅ ∪ ∅ = ∅. Thus, µ(∅) = 2 µ(∅). This is possible only if µ(∅) = 0. It is possible to consider fuzzy hypermeasures in a general case. However, they have definite restrictions. Let us consider a fuzzy hypermeasure µ. For instance, the following results. Lemma 4.2. Any fuzzy hypermeasure is bounded from below. Indeed, for any set B, we have ∅ ⊆ B. This and the property (F4) of fuzzy hypermeasures imply that µ(∅) ≤ µ(B). Corollary 4.1. If µ(∅) = 0, any fuzzy hypermeasure takes only positive values. Hypermeasures have definite algebraic properties. Lemma 4.3. The sum and difference of two (σ-additive, restrictedly σadditive) hypermeasures is also a (σ-additive, restrictedly σ-additive) hypermeasure. Lemma 4.4. If µ is a (σ-additive, restrictedly σ-additive) hypermeasure in X and a is a real number, then aµ is also a (σ-additive, restrictedly σ-additive) hypermeasure in X. Lemmas 4.3 and 4.4 imply the following result. Proposition 4.1. The set M B of all hypermeasures on B is a linear space and the sets M σB of all σ-additive hypermeasures and M rσB of all restrictedly σ-additive hypermeasures on B are a linear subspaces of M B . Lemmas 4.2 and 4.3 imply the following result. + + + + Proposition 4.2. The sets M + B , M f B , M spB , M sbB , M σB , and M + sσB of all positive hypermeasures, positive fuzzy hypermeasures, super-additive, subadditive, σ-additive, and σ -subadditive fuzzy positive hypermeasures on B, respectively, are convex cones in M B .

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Remark 4.6. For linearly positive hypermeasures or linearly positive fuzzy hypermeasures, this result is not true in general as it is possible that the sum of two linearly positive hypermeasures is not a linearly positive hypermeasure. Let (X, B, µ) be a hypermeasure space with the σ-additive hypermeasure µ. Proposition 4.3. If B R is a σ-algebra, then there is a number c ∈ R+ such that for any µ -measurable set A, we have |µ(A)| < c. Corollary 4.2. Any σ-additive measure µ is bounded. Theorem 4.1. For any hypermeasure µ in X defined on a set semiring B, there is a unique extension to a hypermeasure ν defined on the set ring R(B). Proof. For any element A from the set ring R(B), there is a decomposition r ∪i=1 Hi where all Hi ∈ B and Hi ∩ Hj = ∅ when i 6= j [16]. We define ν(A) = Σri=1 µ(Hi ). The value ν(A) does not depend on the choice of the decomposition ∪ri=1 Hi of A. Indeed, if we have two decompositions A = ∪ri=1 Hi and A = ∪ti=1 Ki where all Ki ∈ B and Ki ∩ Kj = ∅ when i 6= j, then all intersections Hi ∩ Kj belong to B as B is a semiring. Thus, Σri=1 µ(Hi ) = Σri=1 Σtj=1 µ(Hi ∩ Kj ) = Σtj=1 µ(Kj ) because the set function is additive. As both decompositions are arbitrary, this proves invariance of the hypermeasure ν. Theorem 4.1 is proved. The general approach to measures and hypermeasures allows one to distinguish positive, linearly positive, negative, and linearly negative measures and hypermeasures. Definition 4.2. A hypermeasure µ in X is called [linearly] negative if all + its values belong to the set R − ω = { α; - α ∈ R ω } [to some linearly ordered − subset of R ω ]. Example 4.1. Let the set D consists of the empty set ∅, R, all closed intervals [a, b] in R, and all sets of the form R \ [a, b]. A set function µ in R is defined as follows: µ(∅) = 0, µ ([a, b]) = |a – b|, µ (R) = Hn (ai )i∈ω , where ai = - i for all i = 1, 2, . . . , and µ (R \ [a, b]) = Hn (ai + |a – b|) i∈ω . By the definitions, µ is a restricted linearly negative hypermeasure in R with the set algebra D. Definition 4.3. A hypermeasure µ in X is called (linearly) restricted if its range R(µ) intersects with a finite number of strict components of the space R ω (and the set of these intersections is linearly ordered by the induced order relation (cf. Section 2) on subsets of R ω ). Example 4.2. Let B = { ∅, [0, 1], R, R \ [0, 1] }. A set function µ in R is defined as follows: µ(∅) = 0, µ ([0, 1]) = 1, µ (R) = Hn (ai )i∈ω where ai = i for all i = 1, 2, . . . , and µ (R \ [0, 1]) = Hn (ai – 1) i∈ω . By the definitions, µ


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is a restricted linearly positive hypermeasure in R with the set algebra B. Lemma 4.5. A measure is bounded if and only if it is restricted as a hypermeasure. Let us find some properties of the space R # ω . As any bounded increasing sequence of real numbers has the limit, we have the following result. Proposition 4.4. Any finite hypernumber α from the space R # ω is a real number. Corollary 4.4. Any component of the space R # ω is its strict component. Let (X, B, µ) be a hypermeasure space. Definition 4.4. A hypermeasure µ is called relatively (linearly) restricted if for any set B from B there is a finite number of bounded sets in the components of the space R ω such that for any A ⊆ B the value µ(A) belongs to one of these bounded sets(and the set of these intersections is linearly ordered by the induced order relation (cf. Section 2) on subsets of R ω ). Lemma 4.6. A measure is relatively bounded [4] if and only if it is relatively restricted as a hypermeasure. Any restricted hypermeasure is relatively restricted and any bounded measure is a restricted hypermeasure. Let (X, B, µ) be a hypermeasure space. Definition 4.5. For any A ∈ B, we define sup µ(A) = sup { µ(D); D ⊆ A and D ∈ B } when this supremum exists. Theorem 4.2. a) If µ is a relatively linearly restricted hypermeasure in X, then sup µ(A) is a linearly positive hypermeasure in X. b) If µ is restrictedly σ-additive (restrictedly σ -subadditive), then sup µ is restrictedly σ-additive (restrictedly σ -subadditive). Proof. a) At first, we show that the function sup µ: B → R ω is defined for all sets A ∈ B. By the conditions of the theorem, µ is a relatively linearly restricted hypermeasure. That is, for an arbitrary set A ∈ B and all µ hypermeasurable subsets C of A, all values µ(C) belong to a finite number of bounded subsets H1 , . . . , Hr of strict components and the set of these subsets is linearly ordered with Hi < Hj for all i < j. It means that any element from Hj is larger than any element from Hj . Consequently, sup µ(A) = sup { µ(D); D ⊆ A and D ∈ B } = sup { µ(D); D ⊆ A, D ∈ B and µ(D) ∈ Hr }. Each Hi is a subset of a strict component SC αi for some hypernumber αi . By Proposition 2.6, SC αi is isomorphic to R as an ordered set. Consequently, for each bounded subset K of the component SC αi , its supremum sup K exists in SC αi . In particular, sup { µ(D); D ⊆ A, D ∈ B and µ(D) ∈ Hr } exists, i.e., the value sup µ(A) is defined. Let A, B, A ∪ B ∈ B and A ∩ B = ∅. Then µ(A ∪ B) = µ(A) + µ(B)

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and for any E ⊆ A and H ⊆ B, we have µ(E ∪ H) = µ(E) + µ(H) because E ∩ H = ∅. Consequently, sup µ(A ∪ B) = sup { µ(D); D ⊆ A ∪ B and D ∈ B } = sup { µ(E ∪ H); E = D ∩ A, H = D ∩ B, D ⊆ A ∪ B and D ∈ B } = sup { µ(E) + µ(H); E = D ∩ A, H = D ∩ B, D ⊆ A ∪ B and D ∈ B } = sup { µ(E); E = D ∩ A, D ⊆ A ∪ B and D ∈ B } + sup { µ(H); H = D ∩ B, D ⊆ A ∪ B and D ∈ B } = sup µ(A) + sup µ(B), i.e., sup µ is an additive mapping of B into R ω . As the mapping sup µ is monotone by its definition and properties of supremum, by Lemma 4.1, sup µ is a positive hypermeasure. Let us show that sup µ is a linearly positive hypermeasure. If we assume that this is not true, then there are sets A and B from B such that neither sup µ(A) ≤ sup µ(B) nor sup µ(A) ≥ sup µ(B). By Theorem 4.1, it is possible to assume that B is a set ring. Consequently, the union C = A ∪ B belongs to B. As is relatively linearly restricted, for all µ -hypermeasurable subsets D of C, all values µ(D) belong to a finite number of bounded subsets H1 , . . . , Hr of strict components and the set of these subsets is linearly ordered with Hi < Hj for all i < j. Then for all µ -hypermeasurable subsets D of C, all values sup µ(D) belong to the closures C H1 , . . . , C Hr of these subsets H1 , . . . , Hr . The union ∪ri=1 Hi is a linearly ordered set. Thus, the union ∪ri=1 C Hi is also a linearly ordered set because all components SC αi are isomorphic to R. Both values sup µ(A) and sup µ(B) belong to the set ∪ri=1 C Hi . Thus, either sup µ(A) ≤ sup µ(B) or sup µ(A) ≥ sup µ(B) is true. This contradicts our assumption and proves that that sup µ is a linearly positive hypermeasure. b) Let us assume that µ is restrictedly σ -subadditive and consider sets A, A1 , A2 , . . . , An , . . . from B R with A ⊆ ∪∞ n=1 An . Taking any E ⊆ A, we have ∞ µ(E) ≤ Σn=1 µ (E ∩ An ) as µ is restrictedly σ -subadditive. By the definition, ∞ µ (E ∩ An ) ≤ sup µ ( An ). Consequently, Σ∞ n=1 µ (E ∩ An ) ≤ Σn=1 sup µ (An ) ∞ for all E ⊆ A. Thus, sup µ (An ) ≤ Σn=1 sup µ (An ), i.e., µ is restrictedly σ -subadditive µ is restrictedly σ -subadditive. Now let us assume that µ is restrictedly σ-additive and consider sets A, A1 , A2 , . . . , An , . . . from B R with A = ∪∞ n=1 An and Ai ∩ Aj = ∅ for all i, j = 1, . . . , m, . . . with i 6= j. Taking any positive number ε, we can find a set Dn ⊆ An , such that sup µ(An ) < µ(Dn ) + (ε\ 2 n ). Then Σ∞ n=1 sup µ ∞ (ε\ 2 n ) = Σ∞ µ(D ) + ε. As the positive number (An ) < Σ∞ µ(D ) + Σ n n n=1 n=1 n=1 ∞ ε is arbitrary, µ is restrictedly σ-additive, and ∪∞ n=1 Dn ⊆ A, we have Σn=1 sup ∞ µ ( An ) ≤ Σ∞ n=1 µ(Dn ) = µ(∪n=1 Dn ) ≤ sup µ (A). At the same time, µ is restrictedly σ -subadditive. So, sup µ (An ) = Σ∞ n=1 sup µ (An ). Theorem 4.2 is proved. Lemma 4.7. If µ is a (relatively) [linearly] restricted hypermeasure in


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X and a is a real number, then aµ is also a (relatively) [linearly] restricted hypermeasure in X. Lemma 4.8. The sum and difference of two (relatively) restricted hypermeasures is also a (relatively) restricted hypermeasure in X. Lemmas 4.7 and 4.8 imply the following result. + Proposition 4.5. The sets M + rB and M rrB of all restricted and relatively restricted hypermeasures on B are convex cones in M B . Lemma 4.9. The sum and difference of two (relatively) σ-additive hypermeasures is also a (relatively) σ-additive hypermeasure in X. Theorem 4.3. (Jordan Decomposition) a) Any relatively linearly restricted hypermeasure µ in X is a difference of two positive hypermeasures. b) If µ is restrictedly σ-additive, then both hypermeasures in this decomposition can be taken restrictedly σ-additive. Proof. a) Let us consider the function π = σ - µ where σ = sup µ. Thus, µ = σ − π. At the same time, by Theorem 4.2, σ is a linearly positive hypermeasure, while Lemma 4.8 and the inequality σ ≥ µ imply that π is also a linearly positive hypermeasure as R # ω is a linear subspace of R ω . b) If µ is restrictedly σ-additive, then it is proved in Theorem 4.2 that σ is also restrictedly σ-additive. By Lemma 4.9, π is also restrictedly σ-additive. Theorem is proved. Corollary 4.5. (see [4]) Any relatively bounded measure µ is a difference of two positive measures. Let us consider additional properties of measures and hypermeasures. Definition 4.6. (see [16]) A σ-additive measure µ in X is called σ -finite if there are µ -measurable sets X1 , X2 , . . . , Xn , . . . such that X = ∪∞ n=1 Xn . Example 4.2. Let X = R, then Lebesgue measure µ in R is σ -finite as we can take the partition P = { Xn = [- n, n]; n = 1, 2, . . . } of R. Definition 4.7. A hypermeasure µ in X is called σ -finite if it is possible to find µ -measurable sets X1 , X2 , . . . , Xn , . . . such that X = ∪∞ n=1 Xn and µ(Dn ) ∈ R for all n = 1, 2, . . . . Let (X, B, µ) be a hypermeasure space with a relatively linearly restricted σ -finite hypermeasure µ. Theorem 4.4. (Hahn Decomposition) There are sets X+ and X− such that X = X+ ∪ X− , X+ ∩ X− = ∅, for any A ⊆ X+ , µ(A) ≥ 0 and for any B ⊆ X− , µ(B) ≤ 0. Proof. a) At first, let us suppose that X ∈ B. As it is proved in Theorem 4.3, µ = σ − π where σ = sup µ and sup µ(A) = sup { µ(D); D ⊆ A and D ∈ B } for any A ⊆ X. In particular, sup µ(X) = sup { µ(D); D ⊆ X and D ∈ B }. Consequently, for any µ –measurable set A, we have σ(A) ≥ µ(A), and

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for any natural number n, there is a set Dn ⊆ X, such that µ(Dn ) > σ(X) – (1 \ 2 n ). Let us define ∞ X+ = lim inf Dn = ∪∞ k=1 ∩n=k Dn

and ∞ X− = X − X+ = lim sup Dn = ∩∞ k=1 ∪n=k (X\Dn ).

Both sets X+ and X− satisfy the conditions of the theorem. To prove this, we show that σ(X− ) = 0 and π(X+ ) = 0. Really, by our construction, σ(Dn ) ≥ µ(Dn ) > σ(X) − (1\2n ). This implies 1\2n > σ(X) − µ(Dn ). Thus, π(Dn ) = σ(Dn ) − µ(Dn ) ≤ σ(X) − µ(Dn ) < 1\2n because by the properties of supremum, σ is a monotone set function. At the same time, σ (X\Dn ) = σ(X) − σ(Dn ) < 1\2n and X− ⊆ ∪∞ n=k (X\D (X\Dn ) for all k = 1, 2, . . . . Consequently, we have σ(X− ) ≤ σ (∪∞ n )) n=k k−1 because σ is a restrictedly σ -subadditive hyperσ ((X\D ) < 1\ 2 ≤ Σ∞ n n=k measure. As k is an arbitrary natural number, we have σ(X− ) =0 Now let us consider π(X+ ). As π is a monotone function, π(Dn ) < 1\ 2 n ∞ ∞ implies that π(∩∞ n=k Dn ) = 0 for all k = 1, 2, . . . . Thus, π(X+ ) = π(∪k=1 ∩n=k ∞ Dn ) ≤ Σ∞ k=1 (∩n=k Dn ) = 0 because by Theorem 4.3, π is restrictedly σ-additive. As a result, we have µ(X+ ) = σ(X+ ) − π(X+ ) = σ(X+ ) ≥ 0 and µ (X− ) = σ(X− ) − π(X− ) = - π(X+ ) ≤ 0. Then for any A ⊆ X+ , µ(A) ≥ 0 and for any B ⊆ X− , µ(B) ≤ 0 because both measures σ and π are positive and monotone. b) Now we consider a general case when X = ∪∞ n=1 Xn and all Xn are µ –measurable. As B is a set algebra, it is possible to take such sets Xn that Xi ∩ Xj = ∅ for all i, j = 1, . . . , m, . . . with i 6= j. The first part of the proof allows us to find for each set Xn , sets Xn+ and Xn− such that Xn = Xn+ ∪ Xn− , Xn+ ∩Xn− = ∅, for any A ⊆ Xn+ , µ(A) ≥ 0 and for any B ⊆ Xn− , µ(B) ≤ 0. ∞ We define X+ = ∪∞ n=1 Xn+ and X− = ∪n=1 Xn− . Taking an arbitrary set A ⊆ ∞ X+ from B, we have µ(A) = µ(∪∞ n=1 A ∩ Xn+ ) = Σn=1 µ(A∩ Xn+ ) ≥ 0 because all µ(A ∩ Xn+ ) ≥ 0 and µ is restrictedly σ-additive. In a similar way, we show that for any B ⊆ X− , µ(B) ≤ 0. Theorem 4.4 is proved. We can induce different partial orders in the set M of hypermeasures in X. Definition 4.8. A hypermeasure µ: B → R ω in X is called an extension of a hypermeasure λ: D → R ω in X if D ⊆ B and λ is a restriction of µ to D. It is denoted by λ ⊆ µ. Definition 4.9. A hypermeasure µ: B → R ω in X is larger than (or equal to) a hypermeasure λ: B → R ω in X if λ(A) < µ(A) (λ(A) ≤ µ(A)) for any A ∈ B. It is denoted by λ < µ (λ ≤ µ).


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We know that σ -finite σ-additive positive measures (for example, the Lebesgue measure) play an important role in the theory of measure and integration. It is natural to introduce σ -finite σ-additive positive hypermeasures in a similar way. However, this does not extend essentially the scope of measurable spaces as the following result demonstrates. Let (X, B, µ) be a hypermeasure space. Theorem 4.5. A σ -finite positive hypermeasure µ is σ-additive and X ∈ B if and only if µ is a σ-additive positive measure and X is µ -measurable. Proof. Sufficiency follows from definitions because (positive) measures are special cases of (positive) hypermeasures. Necessity. Let µ be a σ-additive σ -finite positive hypermeasure µ in X and X ∈ B. It means that X = ∪∞ n=1 Xn , all Xn are µ –measurable, and µ(Xn ) ∈ R. As µ is additive, it is possible to assume that the sequence X1 , X2 , . . . , Xn , . . . is monotone, i.e., Xn ⊆ Xn+1 for all n = 1, 2, . . . . Such sequence X1 , X2 , . . . , Xn , . . . for which X = ∪∞ n=1 Xn is called a partition of X [7]. A partition is cumulative if Xn ⊆ Xn+1 for all n = 1, 2, . . . . Let us consider the sequence of sets Z1 , Z2 , . . . , Zn , . . . where Z1 = X1 and Zn+1 = Xn+1 − Xn for n = 2, 3, . . . . As µ is σ-additive, we have µ (X) = Σ∞ n=1 µ(Zn ) and Zi ∩ Zj = ∅ for i 6= j. By the definition of summation in hypernumbers [7], we have µ(X) = Hn (Σm n=1 µ(Zn )) m∈ω = Hn (µ(Xm )) m∈ω Z . If µ (X) ∈ R ω \ R, µ(X) is a positive as µ is additive and Xm = ∪m n=1 n infinite hypernumber as µ is positive and monotonous. This implies that for any natural number m, there is a natural number n(m) such that µ(Xn(m) ) > m. Let us consider the sequence of numbers { n ([µ (Xk )] + 10 k ) = n [k]; k = 1, 2, . . . } where [µ(Xk )] is the largest natural number smaller than µ (Xk ). Then we can take the partition P = { Xn[k] ; k = 1, 2, 3, . . . } of X. It is cumulative, countable, and µ -measurable because µ is additive. As the hypermeasure µ is σ-additive, we have µ(X) = Σ∞ n=1 µ(Xn[k] ). By the definition of ∞ summation in hypernumbers [7], Σn=1 µ(Xn[k] ) = Hn (Σm n=1 µ(Xn[k] )) m∈ω = Hn (µ(Xn[k] )) m∈ω . However, by properties of hypernumbers, Hn (µ(Xn )) m∈ω 6= Hn (µ(Xn[k] )) m∈ω . This contradiction concludes the proof of the theorem. Corollary 4.6. If a σ -finite positive hypermeasure µ is σ-additive, then µ is a σ-additive positive measure. Proof. If µ is a proper hypermeasure, then there is, at least, one set A ∈ B for which µ(A) is a proper hypernumber. Let us consider a hypermeasure space (A, B A , µA ), where B A = 2 A ∩ B and µA is the restriction of µ to B A . This space satisfies all conditions of Theorem 4.5. Thus, µA is a measure and µ(A) is a number. This contradiction completes the proof.

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Thus, we have extended the scope of application of measure theory by utilization of hypermeasures. This allows one to develop the theory of integration in much more general setting. At the same time, there are alternative approaches to measure theory. For example, one approach to constructing measures suggested by Bourbaki is based on linear forms in vector spaces. Another approach suggested by Federer introduces measures as total functions for all subsets of a measure space, but then restricts measurable sets to a reasonable extent. It would be interesting to extend corresponding constructions of measures for hypermeasures. One more interesting problem is related to the property of linearity for hypermeasures. Many results are obtained for linearly positive, linearly bounded or relatively linearly bounded hypermeasures. The questions are to what extent linearity is essential for these results and what interesting hypermeasures exist that do not satisfy these conditions of linearity. In particular, the proof of Theorem 4.3 shows that in the decomposition µ = σ − π, the hypermeasure σ is linearly positive. Problem. Is it possible to represent any relatively linearly restricted hypermeasure µ in X as a difference of two linearly positive hypermeasures. References [1] J.F. Aarnes, Quasi-states and quasi-measures, Adv. in Math., 86 (1991), 41-67. [2] J.F. Aarnes, A.B. Rustad, Probability and quasi-measures – a new interpretation, Math. Scand., 85 (1999), 278-284. [3] R.B. Ash, Measure, Integration, and Functional Analysis, Academic Press, New York-London (1972). [4] N. Bourbaki, Integration I, Chapters 1-6, Springer Verlag (2004). [5] M. Burgin, Theory of real hypernumbers, In: Properties and Axiomatization, Elsevier, Preprint 0107062 (2001), 42; Electronic Edition: http://www.sciencedirect.com/preprintarchive [6] M. Burgin, Theory of hypernumbers and extrafunctions: functional spaces and differentiation, Discrete Dynamics in Nature and Society, 7, No. 3 (2002), 201-212.


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[7] M. Burgin, Hyperfunctionals and generalized distributions, In: Stochastic Processes and Functional Analysis (Ed-s: A.C. Krinik, R.J. Swift), A Dekker Series of Lecture Notes in Pure and Applied Mathematics, 238 (2004), 81-119. [8] H.A. Cerdeira, Editor, Lectures on Path Integration, Singaporr-Teaneck, World Scientific (1993). [9] Y.L. Dalecky, Y.I. Belopolskaya, Stochastic Equations and Differential Geometry, Vyscha Shkola, Kiev (1989), In Russian. [10] A.Ya. Dorogovtsev, Elements of a General Theory of Measure and Integral, Vyscha Shkola, Kiev (1989), In Russian. [11] R.P. Feynman, A.R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill Companies (1965). [12] D.J. Grubb, Signed quasi-measures, Transactions of the American Mathematical Society, 349, No. 3 (1997), 1081-1089. [13] G.W. Johnson, M.L. Lapidus, The Feynman Integral and Feynman’s Operational Calculus, Oxford Mathematical Monographs, Oxford Univ. Press, Oxford and New York (2002). [14] T. Kashiwa, Path Integral Methods, Oxford Univ. Press, Oxford and New York (1997). [15] G.J. Klir, Z. Wang, Fuzzy Measure Theory, Kluwer Academic Publishers (1993). [16] A.N. Kolmogorov, S.V. Fomin, Elements of Function Theory and Functional Analysis, Nauka, Moscow (1989), In Russian. [17] V. Kolokoltsov, Complex measures on path space: an introduction to the applied to the Schrodinger equation, Methodology and Computing in Applied Probability, 1 (1999), 349-365. [18] V.P. Maslov, Complex Markov Chains and Functional Feynman Integral, Moscow, Nauka (1976), In Russian. [19] M. Oussalah, On the qualitative. Necessity possibility measure, I, Information Sciences, 126 (2000), 205-275.

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[20] M.L. Puri, D. Ralesky, A possibility measure is not a fuzzy measure, Fuzzy Sets and Systems, 7 (1982), 311-313. [21] R.J. Rivers, Path Integral Methods in Quantum Field Theory (Cambridge Monographs on Mathematical Physics), Cambridge Univ Press (1988). [22] K.A. Ross, Elementary Analysis: The Theory of Calculus, Springer, New York-Berlin-Heidelberg (1996). [23] S. Saks, Theory of the Integral, Dover Publications, Inc., New York (1964). ˇ [24] G.E. Silov, B.A. Gurevich, Integral, Measure, and Derivative, Nauka, Moscow (1967), In Russian. [25] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems, 1 (1978), 3-28.