mathematics Article On Four Classical Measure Theorems Salvador López-Alfonso 1 , Manuel López-Pellicer 2,* and Santiago Moll-López 3 1 Department of Architectural Constructions, Universitat Politècnica de València, 46022 Valencia, Spain; [email protected] 2 Emeritus and IUMPA, Universitat Politècnica de València, 46022 Valencia, Spain 3 Department of Applied Mathematics, Universitat Politècnica de València, 46022 Valencia, Spain; [email protected] * Correspondence: [email protected]; Tel.: +34-628-702-360 Abstract: A subset B of an algebra A of subsets of a set W has property (N) if each B-pointwise bounded sequence of the Banach space ba(A) is bounded in ba(A), where ba(A) is the Banach space of real or complex bounded finitely additive measures defined on A endowed with the variation norm. B has property (G) [(VHS)] if for each bounded sequence [if for each sequence] in ba(A) the B- pointwise convergence implies its weak convergence. B has property (sN) [(sG) or (sVHS)] if every increasing covering fBn : n 2 Ng of B contains a set Bp with property (N) [(G) or (VHS)], and B has property (wN) [(wG) or (wVHS)] if every increasing web fBn1n2···nm : ni 2 N, 1 ≤ i ≤ m, m 2 Ng of B contains a strand fBp1 p2···pm : m 2 Ng formed by elements Bp1 p2···pm with property (N) [(G) or (VHS)] for every m 2 N. The classical theorems of Nikodým–Grothendieck, Valdivia, Grothendieck and Vitali–Hahn–Saks say, respectively, that every s-algebra has properties (N), (sN), (G) and (VHS). Valdivia’s theorem was obtained through theorems of barrelled spaces. Recently, it has been proved that every s-algebra has property (wN) and several applications of this strong Nikodým (wN) type property have been provided. In this survey paper we obtain a proof of the property of a s-algebra independent of the theory of locally convex barrelled spaces which depends on elementary Citation: López-Alfonso, S.; basic results of Measure theory and Banach space theory. Moreover we prove that a subset B of an López-Pellicer, M.; Moll-López, S. algebra A has property (wWHS) if and only if B has property (wN) and A has property (G). On Four Classical Measure Theorems. Mathematics 2021, 9, 526. https:// Keywords: algebra and s-algebra of subsets; bounded finitely additive scalar measure; Nikodým; doi.org/10.3390/math9050526 strong and web Nikodým properties; Grothendieck; strong and web Grothendieck properties; Vitali– Hahn–Saks; strong and web Vitali–Hahn–Saks properties Academic Editor: Rekha Srivastava MSC: 28A60; 46G10 Received: 7 February 2021 Accepted: 26 February 2021 Published: 3 March 2021 1. Introduction Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in In this paper A and S denote, respectively, an algebra and a s-algebra of subsets of a published maps and institutional affil- set W. We will refer to an algebra and a s-algebra of subsets of W simply as an algebra and iations. a s-algebra, respectively. The real or complex linear hull L(A) of the set fe(B) : B 2 Ag of characteristics functions of the elements of A with the norm Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. k f k¥ := supfj f (x)j : x 2 Wg, f 2 L(A) This article is an open access article distributed under the terms and is a normed space and its completion is the Banach space L¥(A) of all A-measurable real conditions of the Creative Commons or complex bounded functions defined on W. By [1], Theorem 1.13, its dual endowed with Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Mathematics 2021, 9, 526. https://doi.org/10.3390/math9050526 https://www.mdpi.com/journal/mathematics Mathematics 2021, 9, 526 2 of 17 the polar norm is the Banach space ba(A) of scalar bounded finitely additive measures defined on A, and the polar norm of every m 2 ba(A) is the variation of m, given by n j m j := supfSi=1fj m(Ai)jg : fAi : 1 ≤ i ≤ ng 2 Fg, where F is the family of finite partitions of W by elements of A and m(e(C)) := m(C). For an element B of A the variation of m on B n j m j(B) := supfSi=1fj m(Ai)jg : fAi : 1 ≤ i ≤ ng 2 FBg, defines a seminorm on ba(A) and for each finite partition fBi : Bi 2 A, 1 6 i 6 ng of B we have jmj(B) = Sfjmj(Bi) : 1 6 i 6 ng. The polar set [2], §20, 8 (named absolute polar set), of a subset M of L(A) or ba(A) is the subset M◦ defined by M◦ := fm 2 ba(A) : jm( f )j ≤ 1, for every f 2 Mg, if M ⊂ L(A) or M◦ := f f 2 L(A) : jm( f )j ≤ 1, for every m 2 Mg, if M ⊂ L(A). The topology in ba(A) of pointwise convergence on a subset B of A is denoted by ts(B) in ba(A). Clearly a subset M of ba(A) is ts(B)-bounded if and only if supfjm(C)j : m 2 Mg < ¥, for every C 2 B. ∗ In particular, ts(A) is the weak topology in ba(A). By abscoH is denoted the absolutely convex hull of H and the gauge or Minkowski functional of the subset G := absco(fe(C) : C 2 Ag) of L(A) is a norm in L(A) defined by k f kG := inffjlj : f 2 lGg, f 2 L(A), which is equivalent to the supremum norm ([3], Propositions 1 and 2). Its polar norm in ba(A) is the supremum of the modulus, i.e., for every m 2 ba(A) of m in A, jmj¥ := supfjm(C)j : C 2 Ag, m 2 ba(A), hence in ba(A) the norms variation and supremum are equivalent. For each B 2 A the seminorms defined by the variation on B, jmj(B), and the supremum of the modulus on fC 2 A : C ⊂ Bg, jmj¥,B := supfjm(C)j : C 2 A, C ⊂ Bg, are equivalent seminorms in ba(A). A subset B of an algebra A is a Nikodým set for ba(A), or B has property (N), if the ts(B)-boundedness of a subset M of ba(A) implies supfjmj : m 2 Mg < ¥, or, equivalently supfjm(C)j : m 2 M, C 2 Ag < ¥, i.e., M is uniformly bounded in A. Note that B has property (N) if and only if fe(C) : C 2 Bg is a uniform bounded deciding subset of L(A) ([4], Example 2). We may suppose that the subset M above is absolutely convex and weak*-closed. Clearly B has property (N) if each B-pointwise bounded sequence of ba(A) is uniformly bounded in A. The above set B is a strong Nikodým set for ba(A), or B has property (sN), if each increasing countable covering of B contains an element that has property (N), and B is a web Nikodým set for ba(A), or B has property (wN), if each increasing web fBn1n2···nm : ni 2 N, 1 ≤ i ≤ m, m 2 Ng of B contains a decreasing sequence fBp1 p2···pm : m 2 Ng formed by subsets with property (N). Let us recall that by definition fBn1n2···nm : ni 2 N, 1 ≤ i ≤ m, m 2 Ng is an increasing web of B if the sequence fBn1 : n1 2 Ng is an increasing covering of B and for Mathematics 2021, 9, 526 3 of 17 m each m 2 N and each (n1n2 ··· nm) 2 N the countable family fBn1n2···nmnm+1 : nm+1 2 Ng is an increasing covering of Bn1n2···nm . It is obvious that (wN) ) (sN) ) (N), and that B has property (wN) if and only if each increasing web of B contains an increasing subweb formed by sets that have property (wN). It is straightforward to prove that properties (wN), (s(wN)), (w(sN)) and (w(wN)) are equivalent ([5], Proposition 1). A subset B of an algebra A is a Grothendieck set for ba(A) if each bounded sequence in ba(A) that B-pointwise converges to the null measure converges weakly, and B is a Vitaly-Hahn-Saks set for ba(A) if in ba(A) each sequence that B-pointwise converges to the null measure converges weakly. In brief we will say that B has property (G) or property (VHS), respectively. The above subset B has property (VHS) if and only if B has properties (N) and (G) (see Proposition6). Properties (sG), (wG), (sVHS) and (wVHS) are defined as in the case of properties (sN) and (wN), changing N into G or VHS. For instance, B has property (sG) if for each increasing covering fBn : n 2 Ng of B there exists p 2 N such that Bp has property (G). Clearly (wG) ) (sG) ) (G), and (wVHS) ) (sVHS) ) (VHS). Let us recall (see [6,7])˙ that a subset C of the closed dual unit ball BE∗ of a Banach space E is a Rainwater set for E if for every bounded sequence fxn : n 2 Ng the conditions lim f (xn) = 0, for every f 2 C n!¥ imply ∗ lim f (xn) = 0, for every f 2 E . n!¥ Hence for a subset B of an algebra A the conditions B has property (G) and C = fe(B) : B 2 Bg is a Rainwater set for ba(A) are equivalent ([8], Proposition 4.1) , where e(B) is the element of the closed dual unit ball of ba(A)∗ such that e(B)(m) := m(B), for each measure m 2 ba(A).