On Four Classical Measure Theorems

Salvador López-Alfonso, Manuel López-Pellicer and Santiago Moll-López
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Salvador López-Alfonso: Department of Architectural Constructions, Universitat Politècnica de València, 46022 Valencia, Spain
Manuel López-Pellicer: Emeritus and IUMPA, Universitat Politècnica de València, 46022 Valencia, Spain
Santiago Moll-López: Department of Applied Mathematics, Universitat Politècnica de València, 46022 Valencia, Spain

Mathematics, 2021, vol. 9, issue 5, 1-17

Abstract: A subset B of an algebra A of subsets of a set ? has property ( N ) if each B -pointwise bounded sequence of the Banach space b a ( A ) is bounded in b a ( A ) , where b a ( 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 ) [ ( V H S ) ] if for each bounded sequence [if for each sequence] in b a ( A ) the B -pointwise convergence implies its weak convergence. B has property ( s N ) [ ( s G ) or ( s V H S ) ] if every increasing covering { B n : n ? N } of B contains a set B p with property ( N ) [ ( G ) or ( V H S ) ], and B has property ( w N ) [ ( w G ) or ( w V H S ) ] if every increasing web { B n 1 n 2 ? n m : n i ? N , 1 ? i ? m , m ? N } of B contains a strand { B p 1 p 2 ? p m : m ? N } formed by elements B p 1 p 2 ? p m with property ( N ) [ ( G ) or ( V H S ) ] for every m ? N . The classical theorems of Nikodým–Grothendieck, Valdivia, Grothendieck and Vitali–Hahn–Saks say, respectively, that every ? -algebra has properties ( N ) , ( s N ) , ( G ) and ( V H S ) . Valdivia’s theorem was obtained through theorems of barrelled spaces. Recently, it has been proved that every ? -algebra has property ( w N ) and several applications of this strong Nikodým type property have been provided. In this survey paper we obtain a proof of the property ( w N ) of a ? -algebra independent of the theory of locally convex barrelled spaces which depends on elementary basic results of Measure theory and Banach space theory. Moreover we prove that a subset B of an algebra A has property ( w W H S ) if and only if B has property ( w N ) and A has property ( G ) .

Keywords: algebra and ?-algebra of subsets; bounded finitely additive scalar measure; Nikodým; strong and web Nikodým properties; Grothendieck; strong and web Grothendieck properties; Vitali–Hahn–Saks; strong and web Vitali–Hahn–Saks properties (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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