Mathematica Aeterna, Vol. 7, 2017, no. 3, 287 - 293
Family of Strongly Additive Vector Measures∗
Xin-Lei Yong, Yi-Fan Han, Ling-Shan Xu, Yuan-Hong Tao† Department of Mathematics, College of Science, Yanbian University, Jilin, 133002, China
Abstract
The family of strongly additive vector measures is characterized in this paper. Firstly, the sufficient and necessary condition of a vector measure, which takes values in a completely Hausdorff topological vector space, to be strongly addtive is established. Then BTB spaces are discussed and a Diestel-Faires type result is obtained.
Mathematics Subject Classification: Primary 46A03, 46E40
Keywords:vector measures; strongly additive measure, bounded measure, totally bounded set, precompact set
1 Introduction
Vector measures have long been of interest to measure theorists, the general theory can be found in [1-4]. In recent years extensive work has been done on the additivity of vector measures ([5-14]). Olav Nygaard, M¨art P˜oldvere ([11]) considered vector measures that take values in Banach spaces, they character- ized families of vector measures of uniformly bounded variation and semivari- ation in terms of additivity properties, and simplified the proof of Nikodyms boundedness theorem. Throughout this paper, F will be a field (i.e.,algebra) of subsets of a set and G an abelian topological group with the family N (G) of neighborhoods 0 ∈ G. A net (xα)α∈(I,≤) in G is Cauchy if for every U ∈ N (G) there is an α0 ∈ I such that xα − xβ ∈ U whenever α ≥ α0 and β ≥ α0. G is complete if every Cauchy net in G is convergent. For a finitely additive measure µ : F → G, the most important event is ∞ the behavior of the sequence {µ(Aj)}j=1 where {Aj} is pairwise disjoint. In
∗Supported by Natural Science Foundation of China (11361065,11761073) †Corresponding Author: Tao Yuanhong ( 1973 -), female, Yanbian University, Depart- ment of mathematics, Ph.D., Associate Professor, Major in functional analysis and its ap- plication. E-mail: [email protected] 288 Xin-Lei Yong, Yi-Fan Han, Ling-Shan Xu, Yuan-Hong Tao
∞ fact, µ : F → G is strongly additive if Pj=1 µ(Aj) converges whenever {Aj} is pairwise disjoint ([3], P.7). For Banach space X with dual X′, a result in [11] ∞ says that µ : F → X is bounded variation if and only if Pj=1 µ(Aj) < +∞ whenever {Aj} is pairwise disjoint ([11], Cor.2), and µ : F → X is bounded, i.e., µ(F) is bounded ([3], P.4) if and only if whenever {Aj} is pairwise disjoint ∞ ′ ′ in F, then Pj=1 |x (µ(Aj))| < +∞ for each continuous linear functional x ∈ X′ ([11], Cor.4). In this paper, we would like to characterize the family of strongly additive vector measures in terms of additivity property. As an application, we will show a Diestel-Faires type result ([3], P.20,Theorem 2; [2]) which is also an important fact in analysis.
2 Strongly Additive Vector Measures
Definition 2.1 For {xj} ⊂ G and △ = {j1, j2, } ⊆ N with j1 < j2 < , let ∞
X xj = X xjk j∈△ k=1 and Pj∈△ xj =0 if △ = φ. Ronglu Li, Hao Guo and C. Swartz ([16]) showed that every abelian topo- logical group contains many interesting sets which are both compact and se- quentially compact, they also deduced some useful facts: Theorem A ([16, Theorem 1]). Let Ω be a compact (resp., sequentially compact) space and G an abelian topological group. If {fj} ⊂ C(Ω,G) is ∞ ∞ such that Pj=1 fj(ωj) converges for each {ωj} ⊂ Ω, then {Pj=1 fj(ωj): ωj ∈ Ω, ∀j ∈ N} is compact (resp., sequentially compact). Theorem B ([16, Corollary 2]). Let G be an abelian topological group and {xj} ⊂ G. If Pj xj is subseries convergent, i.e., Pj∈△ xj converges for each △ ⊂ N, then the set {Pj∈△ xj : △ ⊂ N} is both compact and sequentially compact. Theorem C ([16, Theorem 2]). Let G be an abelian topological group. Then for every countably additive µ :2N → G, the range µ(2N )= {µ(A): A ⊂ N} is both compact and sequentially compact. Moreover, if P is a σ -algebra of subsets of a set Ω and µ : P → G is countably additive, then for every pairwise disjoint {Aj} ⊂ P, the set {Pj∈△ µ(Aj): △ ⊂ N} is both compact and sequentially compact. Similar to the case of Banach spaces, we have the following simple fact. Lemma 1. Let G be a Hausdorff abelian topological group and µ : F → G be a vector measure. If G is complete, then the following four results are equivalent. (1) µ is strongly additive; Family of Strongly Additive Vector Measures 289
(2) If {Aj} is pairwise disjoint in F, then limj µ(Aj)=0; (3) If A1 ⊆ A2 ⊆ in F, then limj µ(Aj) exists; (4) If A1 ⊇ A2 ⊇ in F, then limj µ(Aj) exists. Theorem 1. Let X be a Hausdorff topological vector space and µ : F → X be a vector measure. If X is complete, then µ is strongly additive if and only if {Pj∈△ µ(Aj): finite △ ⊂ N} is totally bounded (i.e., precompact, [15], P.83) for every pairwise disjoint sequence {Aj}⊂F. Proof. (⇒). Let {Aj} be pairwise disjoint in F and B = {Pj∈△ µ(Aj): finite △ ⊂ N}. For each △ = {j1, j2, }⊆ N with j1 < j2 < , we can ∞ easily get that {Ajk } is pairwise disjoint and j∈△ µ(Aj)= k=1 µ(Ajk ) con- ∞P P verges by the strong additivity of µ. Thus, Pj=1 µ(Aj) is subseries convergent, i.e., Pj∈△ µ(Aj) converges for each △ ⊆ N. Let S = {Pj∈△ µ(Aj): △ ⊆ N}. By Theorem B, S is both compact and sequentially compact in X. Then B is totally bounded since B ⊂ S. n ∞ (⇐). Let {Aj} be pairwise disjoint in F. If {Pj=1 µ(Aj)}n=1 is not Cauchy, then we have a balanced U ∈N (X) and integer sequence m1 ≤ n1 < m2 ≤ n2 < such that nk xk = X µ(Aj) ∈/ U j=mk for all k. Then pick a balanced V ∈ N (X) such that V + V ⊂ U. Since B = {Pj∈△ µ(Aj): finite △ ⊂ N} is totally bounded, B ⊂ pV for some p ∈ N. Then pick a balanced W ∈N (X) for which
(p) zW + W}| + W{ ⊂ V.
Since {xk} ⊂ B, then {xk : k ∈ N} is totally bounded, and there is a finite △ ⊂ N such that {x1, x2, x3, }⊂{xk : k ∈ △}+W ([15], P.86, Prob.6) and so there is a k0 ∈△ such that xk0 +W contains infinite vectors in {xk : k ∈ N}. ∞ Say that {xki }i=1 ⊂ xk0 + W . Then p p ( )
X xki ∈ (zxk0 + W )+(xk0 + W}| )+ +(xk0 + W{) i=1 (p)
= pxk0 + Wz + W +}| + W{
⊂ pxk0 + V. p Hence Pi=1 xki = pxk0 + v for some v ∈ V . However, V = −V and p p 1 1 x = ( x − v) ∈ ( x − V ) k0 p X ki p X ki i=1 i=1 290 Xin-Lei Yong, Yi-Fan Han, Ling-Shan Xu, Yuan-Hong Tao while
p p 1 1 ( x − V ) = ( x + V ) p X ki p X ki i=1 i=1 p nki 1 = [ µ(A )+ V ] p X X j i=1 j=mki 1 ⊂ (B + V ) p 1 ⊂ (pV + V ) p 1 = V + V p ⊂ V + V ⊂ U.
n ∞ This contradicts that xk ∈/ U for all k and so { j=1 µ(Aj)}n=1 is Cauchy in ∞ P n X. Since X is complete then Pj=1 µ(Aj) = limn Pj=1 µ(Aj) exists.
3 Diestel-Faires Type Result
A topological vector space X is called a BT B space if every bounded set in X is totally bounded ([15], P.85). Many important spaces are BT B spaces, e.g., Rn,Cn, RN ,CN (= ω), the space D of test functions, semi-Montel spaces ([15], P.90), etc. In fact, we have many BT B spaces as follows.
Lemma 3.1 Let X be a complete Hausdorff BT B space and XΩ all the mappings from Ω to X. For every Ω = φ, let σΩ be the topology for XΩ such Ω Ω that fα → f in (X , σΩ) if and only if fα(ω) → f(ω), ∀ω ∈ Ω. Then (X , σΩ) is a complete Hausdorff BT B space. Proof. Each ω ∈ Ω gives a function ω : XΩ → X such that ω(f)= f(ω), ∀f ∈ XΩ. Letting ω−1(φ)= φ and denoting σω = {ω−1(G): G is open in X} be a topology for XΩ and σΩ= sup{σω : ω ∈ Ω} ([15], P.11). Ω If f,g ∈ X , f = g, then f(ω0) = g(ω0) for some ω0 ∈ Ω and so [f(ω0)+ −1 U] ∩ [g(ω0) + U] = φ for some open U ∈ N (X). Then ω0 [f(ω0) + U] ∩ −1 Ω ω0 [g(ω0)+ U]= φ and so (X , σΩ) is Hausdorff. Ω Let (fα)α∈(I,≤) be a Cauchy net in (X , σΩ). Then (fα)α∈I is Cauchy in (XΩ,σω) for each ω ∈ Ω ([15], P76, Prob.4). Fix an ω ∈ Ω. For every open V ∈ N (X),ω−1(V ) = {f ∈ XΩ : f(ω) = ω(f) ∈ V } is an open neighborhood ω −1 of 0 ∈ (X ,σω) and so there is α0 ∈ I such that fα − fβ ∈ ω (V ) for all α, β ≥ α0, i.e., fα(ω) − fβ(ω)=(fα − fβ)(ω) = ω(fα − fβ) ∈ V for all α, β ≥ α0. Then (fα(ω))α∈I is Cauchy in X and so limα fα(ω)= f(ω) exists. Family of Strongly Additive Vector Measures 291
Ω Ω Ω Thus, we have a f ∈ X such that fα → f in (X , σΩ), i.e., (X , σΩ) is complete. σΩ Let S be a bounded set in (XΩ, σΩ) and S be the closure of S in (XΩ, σΩ) σΩ 1 σΩ 1 which is also bounded. If {fn} ⊂ S , then n fn → 0 and so n fn(ω) → 0 for σΩ all ω ∈ Ω, i.e., {f(ω): f ∈ S } is bounded in X for each ω ∈ Ω. Since X σΩ is a complete BT B space, {f(ω): f ∈ S } is totally bounded and complete σΩ in X for each ω ∈ Ω, i.e., {f(ω): f ∈ S } is compact for each ω ∈ Ω ([15], σΩ P.88, Th.7). Then S is compact in (XΩ, σΩ) ([17], P218, Th.1) and so S is totally bounded in (XΩ, σΩ). A very nice Diestel-Faires theorem shows that a Banach space X contains no copy of c0 if and only if every bounded measure µ : F → X is strongly additive ([3], P.20, Th.2; [2]). But Lemma 2 shows that the family of BT B spaces includes many of non-metrizable spaces and so we have a nice fact as follows.
Theorem 3.2 Let X be a complete Hausdorff BT B space. If µ : F → X is a bounded measure, i.e., µ(F) = {µ(A): A ∈ F} is bounded in X , then µ is strongly additive and for every pairwise disjoint sequence {Aj} ⊂ F , {Pj∈△ µ(Aj): △ ⊆ N} is both compact and sequentially compact. Proof. Let {Aj} be pairwise disjoint in F. Then
K = {X µ(Aj): finite △⊆N} = {µ(∪j∈△Aj): finite △⊆N}⊂ µ(F) j∈△ is bounded in X and so K is totally bounded. By Theorem 1, µ is strongly additive. As in the proof of Theorem A and Theorem B implies that {Pj∈△ µ(Aj): △⊆N} is both compact and sequentially compact whenever {Aj} is pairwise disjoint in F. Theorem C says that if µ : 2N → G is countably additive, then the range µ(2N ) = {µ(A): A ⊆ N} is both compact and sequentially compact . For strongly additive measures we also have a similar result as follows.
Corollary 3.3 Let X be a Hausdorff topological vector space.If X is com- N plete and µ : 2 → X is a strongly additive measure, then {Pj∈A µ(j): A ⊆ N} is both compact and sequentially compact. Proof. If i = j in N , then {i}∩{j} = φ and A = ∪j∈A{j} whenever A ⊆ N . Then Pj∈A µ({j}) converges for each A ⊆ N . As in the proof of Theorem 1, the desired result follows from Theorem B.
ACKNOWLEDGEMENTS. In this paper, we first disscussed the fam- ily of strongly additive vector measures. We proved that a vector measure 292 Xin-Lei Yong, Yi-Fan Han, Ling-Shan Xu, Yuan-Hong Tao
µ : F → X, where X is a complete Hausdorff topological vector space, strongly additive if and only if {Pj∈△ µ(Aj): finite △ ⊂ N} is totally bounded for every pairwise disjoint sequence {Aj}⊂F. We also discussed the BT B space, and established a Diestel-Faires type theorem: If X be a complete Hausdorff BT B space. and µ : F → X is a bounded measure, then µ is strongly additive and for every pairwise disjoint sequence {Aj} ⊂ F , {Pj∈△ µ(Aj): △ ⊆ N} is both compact and sequentially compact.
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Received: August 10, 2017