Vitali-Hahn-Saks Property in Coverings of Sets Algebras

Abstract

[EN] A subset B of an algebra A of subsets of Omega is a Nikodym set for ba(A) if each B-pointwise bounded subset M of ba(A) is uniformly bounded on A and B is a strong Nikodym set for ba(A) if each increasing covering (B-m)(m=1)(infinity) of B contains a B-n which is a Nikodym set for ba(A), where ba(A) is the Banach space of the real (or complex) finitely additive measures of bounded variation defined on A. The subset B has (VHS) property if B is a Nikodym set for ba(A) and for each sequence (mu(n))(m=1)(infinity) and each mu, both in ba(A) and such that lim(n -> 8) mu(n)(B) = mu(B), for each B is an element of B, we have that the sequence (mu(n))(m=1)(infinity) converges weakly to mu. We prove that if (B-m)(m=1)(infinity) is an increasing covering of and algebra A that has (VHS) property and there exist a B-n which is a Nikodym set for ba( A) then there exists B-q, with q >= p, such that B-q has (VHS) property. In particular, if (B-m)(m=1)(infinity) is an increasing covering of a sigma-algebra there exists B-q that has (VHS) property. Valdivia proved that every sigma-algebra has strong Nikodym property and in 2013 asked if Nikodym property in an algebra implies strong Nikodym property. We present three open questions related with this aforementioned Valdivia question and a proof of his strong Nikodym Theorem for sigma-algebras that it is independent of the Barrelled spaces theory and it is developed with basic results of Measure theory and Banach spaces.