On Four Classical Measure Theorems

A subset B of an algebra subsets a set Ω has property (N) if each B-pointwise bounded sequence the Banach space ba(A) is in ba(A), where real or complex finitely additive measures defined on endowed with variation norm. (G) [(VHS)] for [if sequence] convergence implies its weak convergence. (sN) [(sG) (sVHS)] every increasing covering {Bn:n∈N} contains Bp [(G) (VHS)], and (wN) [(wG) (wVHS)] web {Bn1n2⋯nm:ni∈N,1≤i≤m,m∈N} strand {Bp1p2⋯pm:m∈N} formed by elements Bp1p2⋯pm (VHS)] m∈N. The classical theorems Nikodým–Grothendieck, Valdivia, Grothendieck Vitali–Hahn–Saks say, respectively, that σ-algebra properties (N), (sN), (VHS). Valdivia’s theorem was obtained through barrelled spaces. Recently, it been proved several applications this strong Nikodým type have provided. In survey paper we obtain proof independent theory locally convex spaces which depends elementary basic results Measure theory. Moreover prove (wWHS) only (G).