On Four Classical Measure Theorems

A subset B of an algebra A of subsets of a set Ω 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 σ -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 σ -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 σ -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 ) .