Georgian Math. J. ????; ??? (???):1–5
Research Article Sanjib Basu* and Krishnendu Dutta
A decomposition theorem for Locally Compact groups DOI: 10.1515/gmj-XXXX, Received April 4, 2015; accepted June 29, 2015
Abstract: Here we prove that, under certain restrictions, every locally compact group equipped with a nonzero, σ-finite, regular left Haar measure can be decomposed into two small sets, one of which is small in the sense of measure and the other is small in the sense of category, and all such decompositions originate from a generalized notion of a Lebesgue point. Incidentally, such class of topological groups for which this happens turns out to be metrizable. We also observe an interesting connection between Luzin sets in such spaces and decompositions of the above type. Keywords: First Category sets, σ-finite Borel measure, measure zero cardinal, locally compact Hausdorff, Vitali system, admissible system of sets, generalised Lebesgue point, Hardy–Littlewood inequality, Luzin set MSC 2010: Primary 28A; Secondary 28C15
1 Introduction There is a well known decomposition theorem [9] which states that the real line can be expressed as the disjoint union of a Lebesgue null set and a set of first category. The result is significant for although the real line is neither a set of first category nor a set of measure zero (which means that it is small neither in the sense of measure nor in the sense of category), each of two components in whose disjoint union it can be expressed is small in one sense but not in the other. Hence we refer to some generalizations of the above theorem. Theorem 1.1 ([9]). If X is a separable metric space, then a decomposition of X into a µ-null set and a set of first category exists for each σ-finite diffused Borel measure µ on X. However, separability and σ-finiteness may be replaced by comparatively less stringent requirements. We say that a cardinal is of measure zero if every finite non-atomic (or diffused) measure defined for all subsets of any set of that cardinality vanishes identically [10]. Any cardinal less than one of measure zero is of measure zero and also all cardinals less than the first weakly inaccessible cardinal are of measure zero. This concept of a measure zero cardinal has been used earlier to derive many interesting results such as the measure analogue of the Banach category theorem [10] and also an invariant extension of the Lebesgue measure [5]. A measure µ is called semifinite if every set of infinite measure contains a set of positive finite measure. With these two definitions, we now state the following generalization of the above theorem. Theorem 1.2 ([10]). Let X be a metric space having a base of measure zero cardinal, and µ be a diffused (or non-atomic) Borel measure on X such that (i) µ is semifinite, (ii) every set of measure zero is contained in a Gδ set of measure zero. *Corresponding Author: Sanjib Basu: Department of Mathematics, Bethune College, 181 Bidhan Sarani, Kolkata-700 006, W.B., India, e-mail:
[email protected] Krishnendu Dutta: Govt. College of Engg. & Ceramic Technology, 73, A. C. Banerjee Lane, Kolkata, India, e-mail:
[email protected]
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S. Basu and K. Dutta, A decomposition theorem for Locally Compact groups
Then X can be expressed as the disjoint union of a set of measure zero and a set of first category. The above theorem does not hold for general topological spaces. For there exist a normal bicompact and totally disconnected space and a finite Borel measure µ on it such that µ(E) = 0 if and only if E is a set of first category [9]. But Theorem 1.3 ([2]). In a topological space X having a countable dense subset Y and whose every oneelement set {x} is Gδ , there exists a Gδ set E ⊇ Y such that µ(E) = 0 for every σ finite, non-atomic Borel measure µ on X. In this theorem, the complement of E in X is certainly a first category set. So, any separable topological space subject to such restrictions as stated above admits a decomposition into a µ-null set and a set of first category. Since in any locally compact Hausdorff space, first countability implies that every one-element set {x} is Gδ , every locally compact Hausdorff first countable space satisfies the conditions of the theorem. Such spaces are not necessarily metrizable, as may be observed from the example of Helley space [6], which constitutes the family of all nondecreasing functions from the unit interval [0, 1] into itself with the topology induced by the product topology on [0, 1][0,1] .
2 Preliminaries and results Throughout this paper, G denotes a locally compact Hausdorff topological group with e as the identity element, S1 is the σ-ring generated by compact subsets of G [3], S the σ-ring generated by S1 and subsets of sets in S1 of µ-measure zero, where µ is a non zero, σ-finite, diffused (this property is equivalent to the nondiscreetness of the group) and a regular left Haar measure on S. The outer measure of any set E ⊆ G induced by µ is given by µ∗ (E) = inf{µ(F ) : E ⊆ F ∈ S}. The outer measure µ∗ is σ-finite on E ∗ if E ⊆ ∪∞ use the n=1 En where µ (En ) < ∞ for each n. It is called totally σ-finite if it is σ-finite on G. We R 1 standard notation L (G) for the class of all real-valued µ-integrable functions on G for which G f dµ is R finite. The class L1 (G) is a topological space under the standard norm ||f ||1 = G |f |dµ. It is also supposed that our topological group G is endowed with a system of sets which we call the “Vitali system” and whose definition reads as follows. Definition 2.1. A system (or a class) C of compact subsets of G is called a Vitali system if it has the following set of properties: (i) e ∈ S and µ(S) > 0 for every S ∈ C; (ii) for every open neighbourhood V of the identity element e, there is g ∈ G and S ∈ C such that e ∈ gS ⊆ V ; (iii) for every sequence {gn Sn }∞ n=1 satisfying e ∈ gn Sn where gn ∈ G, Sn ∈ C and limn→∞ µ(Sn ) = 0 and every open neighbourhood V of e, we have e ∈ gn Sn ⊆ V for sufficiently large n, and (iv) if the outer measure µ∗ is σ-finite on A and there exists a subclass C ∗ of C satisfying condition (i) and ∗ (ii), then there exists a sequence {gn Sn }∞ n=1 (gn ∈ G, Sn ∈ C ) such that gn Sn are mutually disjoint and µ∗ (A \ ∪∞ n=1 gn Sn ) = 0. Remark 2.2. Since from the nondiscreetness of the group G, it follows that for each n there exists an open set Vn such that x ∈ Vn with µ(Vn ) < n1 , condition (ii) of Definition 2.1 ensures that to each x ∈ G there corresponds a sequence {gn Sn }∞ n=1 (gn ∈ G, Sn ∈ C) such that x ∈ gn Sn for each n and limn→∞ µ(Sn ) = 0. Thus in the present context, condition (iii) becomes meaningful. It may also be noted that conditions (ii) and (iii) are equivalent to the following ones: (ii)0 if x ∈ V (open), there exists g ∈ G and S ∈ C such that x ∈ gS ⊆ V ; (iii)0 for every sequence {gn Sn }∞ n=1 satisfying x ∈ gn Sn for each n and limn→∞ µ(Sn ) = 0, if x ∈ V (open), then gn Sn ⊆ V for all n sufficiently large.
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S. Basu and K. Dutta, A decomposition theorem for Locally Compact groups
If instead of satisfying all the four conditions of Definition 2.1 which characterises the Vitali system, the class C satisfies only the first three of them, then it is called an admissible class [1] and with respect to such a class of sets, we define the notion of a generalized Lebesgue point or C-point. Definition 2.3 ([1]). A point x ∈ G is called a C-point of a function f ∈ L1 (G) provided that R |f (y) − f (x)|dµ gS lim sup = 0, µ(S) gS→x R R n
where lim supgS→x
gS
|f (y)−f (x)|dµ µ(S)
expresses the quantity sup
lim supn→∞
gn S n
|f (y)−f (x)|dµ o µ(Sn )
, the
supremum being taken over all sequences {gn Sn }∞ n=1 (gn ∈ G, Sn ∈ C) such that x ∈ gn Sn for all n and limn→∞ µ(Sn ) = 0. Since the Euclidean n-space with its usual topology and group structure is a locally compact Hausdorff topological group where with respect to the usual topology and group structure, the family of closed balls centered at the origin forms an admissible class, the notion of a C-point is an extension of the classical notion of a Lebesgue point. Moreover, the above definition depends on the choice of an admissible class. Also, as in [1], as a natural generalization of Lebesgue set, we denote by C(f ) the set of all C-points of any f ∈ L1 (G), i.e., C(f ) = {x ∈ G : x is a C-point of f }. However, from now on throughout the paper, C will continue to represent the Vitali system instead of an admissible class of sets. We intend to prove in this paper that there is a residual class of functions in L1 (G) with the property that each function in that class gives rise to a decomposition of G into a µ∗ -null set and a set of first category. To prove this, we proceed in steps. We first establish the following generalization of a very well known classical theorem by Lebesgue. We assume here that µ∗ is totally σ-finite on G. Proposition 2.4. For every f ∈ L1 (G), µ∗ -almost every point of G is a C-point of f ; in symbols µ∗ (G \ C(f )) = 0. A proof of the above proposition depends on the following lemma which in the present context is an appropriate variation of the Hardy–Littlewood inequality. R |f |dµ
Lemma 2.5.R For any f ∈ L1 (G) and t > 0, µ∗ {x ∈ G : f ∗ (x) > t} < G t where f ∗ (x) = n| o f (y)dµ| gS inf V sup : x ∈ gS where g ∈ G, S ∈ C such that gS ⊆ V , the infimum being taken over µ(S) all neighbourhoods V of x. Proof. Let A = {x ∈ G : f ∗ (x) > t}. Then according to the definition of f ∗ , for each x ∈ G and each neighbourhood Vx of x, there exist gx ∈ G and Sx ∈ C such that x ∈ gx Sx ⊆ Vx and µ(Sx ) < R 1 ∗ t | gx Sx f dµ|. Now since {Sx : x ∈ A} can play the role of C in condition (iv) of Definition 2.1, we may ∞ ∗ choose a sequence {gxn Sxn }n=1 (gxn ∈ G, Sxn ∈ C ) such that the sets gxn Sxn are mutually disjoint and R P∞ P∞ R 1 1 ∗ µ∗ (A \ ∪∞ n=1 gxn Sxn ) = 0. Hence µ (A) ≤ n=1 µ(gxn Sxn ) ≤ t n=1 gxn Sxn |f |dµ < t G |f |dµ, which proves the lemma. 1 Proof of Proposition 2.4. We choose = ij where i, j are positive integers. Let h be a continuous function R 1 with compact support such that G |f − h|dµ < ij . We define two subsets E and F of G as follows: 1 ∗ E = {x ∈ G : |f − h| (x) > i } and F = {x ∈ G : |f (x) − h(x)| ≥ 1i }. By Lemma 2.5, µ∗ (E) < 1j and also R 1 since 1i µ∗ (F ) ≤ F |f (x) − h(x)|dµ ≤ ij , we have µ∗ (F ) < 1j . Also for any g ∈ G and a compact set S R R R containing e we can write gS |f (y)−f (x)|dµ ≤ gS |f (y)−h(y)|dµ+ gS |h(y)−h(x)|dµ+µ(gS)|h(x)−f (x)|.
Let δ > 0 be chosen arbitrarily. Then the complement of E in G is contained in the set E (δ) = {x ∈ G : |f −h|∗ (x) ≤ 1i +δ} and so from the definition of the ∗-mapping given above and the continuity of h it follows R (δ) 1 that whenever x 6∈ E ∪F there exists an open neighbourhood Vx of x such that µ(S) |f (y)−h(y)|dµ < gS 1 i
(δ)
(δ)
+ δ for every g ∈ G and S ∈ C such that x ∈ gS ⊆ Vx . Moreover, |h(y) − h(x)| < δ for every y ∈ Vx . Now let {gn Sn }∞ n=1 ( gn ∈ G, Sn ∈ C) be a sequence satisfying x ∈ gn Sn for every n and (δ) limn→∞ µ(Sn ) = 0. Then according to condition (iii)0 (see Remark 2.2), gn Sn ⊆ Vx for all n after a
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S. Basu and K. Dutta, A decomposition theorem for Locally Compact groups
R certain stage onwards, and hence for all such gn Sn we may write R (δ)
gn Sn
|f (y)−f (x)|dµ
µ(Sn ) |f (y)−f (x)|dµ
