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 , 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 , which takes values in a completely Hausdorff topological , 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 {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 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 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 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 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.

References

[1] N. Dinculeanu, Vector Measures, VEB Deutscher Verlag der Wis- senschaften, Berlin, 1966.

[2] J. Diestel and B.Faires. On Vector Measures. Trans. Amer. Math. Soc. Vol.198, pp. 253-271, 1974.

[3] J. Diestel and J.J. Uhl, Jr., Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence, 1977.

[4] J. Diestel and J.J. Uhl, Jr., Progress in Vector Measures: 197783, Lecture Notes Math. 1033, Springer, Berlin Heidelberg, 1984, pp. 144-192.

[5] Jan Hamhalter, Additivity of Vector Gleason Measures, International Journal of Theoretical Physics, Vol. 31, No. 1, 1992.

[6] Helmut H. Schaefer and Xiaodong Zhang, A Note on Order Bounded Vector Measures, Arch. Math., Vol. 63, pp. 152-157 , 1994.

[7] A. G. Areshkina, Fourier of Additive Vector Measures and Their Term-by-Term Differentiation,Mathematical Notes, Vol. 64, No. 2, 1998.

[8] M. C. Romero-Moreno, The Range of a Vector Measure and a Radon- Nikodym Problem for the variation, Arch. Math. Vol.70, pp. 74-82, 1998.

[9] A. I. Sotnikov, Order Properties of the Space of Strongly Additive Transi- tion Functions, Siberian Mathematical Journal, Vol. 46, No. 1, pp. 166171, 2005.

[10] A. G. Chentsov, Finitely Additive Measures and Extensions of Abstract Control Problems, Journal of Mathematical Sciences, Vol. 133, No. 2, pp. 1045-1206, 2006.

[11] Olav Nygaard, M¨art P˜oldvere. Families of Vector Measures of Uniformly Bounded Variation. Arch. Math. Vol. 88, pp. 57-61, 2007. Family of Strongly Additive Vector Measures 293

[12] G.P. Curbera and W.J. Ricker, Vector Measures, Integration and Appli- cations, Positivity Trends in Mathematics, PP. 127-160, 2007.

[13] Diomedes Barcenas and Carlos E. Finol, On Vector Measures, Uniform Integrability and Orlicz Spaces,Operator Theory: Advances and Applica- tions, Vol. 201, pp. 5157, 2009.

[14] Beloslav Riecan, On The Kluvanek Construction of the lebesgue Integral with Respect to a Vector Measure, Math. Slovaca, Vol. 64, No. 3, pp. 727740, 2014.

[15] A.Wilansky. Modern Methods in Topological Vector Spaces. McGraw-Hill. New York, 1978.

[16] Ronglu Li and C.Swartz. Both Compact and Sequentially Compact Sets in Abelian Topological Group. Topology and its Applications. Vol. 158, pp. 1234-1238, 2011.

[17] J.L.Kelley. General Topology. Van Nostrand, 1955.

Received: August 10, 2017