On Four Classical Measure Theorems

Abstract

[EN] A subset B of an algebra A of subsets of a set Omega 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 {B-n:n is an element of N} of B contains a set B-p with property (N) [(G) or (VHS)], and B has property (wN) [(wG) or (wVHS)] if every increasing web {Bn(1)n(2)...n(m) : n(i) is an element of N,1 <= i <= m,m is an element of N} of B contains a strand {B-p1p2...pm: m is an element of N} formed by elements B-p1p2...pm with property (N) [(G) or (VHS)] for every m is an element of N. The classical theorems of Nikodym-Grothendieck, Valdivia, Grothendieck and Vitali-Hahn-Saks say, respectively, that every sigma-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 sigma-algebra has property (wN) and several applications of this strong Nikodym type property have been provided. In this survey paper we obtain a proof of the property (wN) of a sigma-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 (wWHS) if and only if B has property (wN) and A has property (G).