Vitali–Hahn–Saks property in coverings of sets algebras

Abstract

A subset $\mathscr {B}$ of an algebra $\mathscr {A}$ of subsets of $\varOmega$ is a Nikodým set for $ba(\mathscr {A})$ if each $\mathscr {B}$-pointwise bounded subset M of $ba(\mathscr {A})$ is uniformly bounded on $\mathscr {A}$ and $\mathscr {B}$ is a strong Nikodým set for $ba(\mathscr {A})$ if each increasing covering $(\mathscr {B}_{m})_{m=1}^{\infty }$ of $\mathscr {B}$ contains a $\mathscr {B}_{n}$ which is a Nikodým set for $ba(\mathscr {A})$, where $ba(\mathscr {A})$ is the Banach space of the real (or complex) finitely additive measures of bounded variation defined on $\mathscr {A}$. The subset $\mathscr {B}$ has (VHS) property if $\mathscr {B}$ is a Nikodým set for $ba(\mathscr {A})$ and for each sequence $(\mu _{n})_{m=1}^{\infty }$ and each $\mu$, both in $ba(\mathscr {A})$ and such that $\lim _{n\rightarrow \infty }\mu _{n}(B)=\mu (B)$, for each $B\in \mathscr {B}$, we have that the sequence $(\mu _{n})_{m=1}^{\infty }$ converges weakly to $\mu$. We prove that if $(\mathscr {B} _{m})_{m=1}^{\infty }$ is an increasing covering of and algebra $\mathscr {A}$ that has (VHS) property and there exist a $\mathscr {B}_{n}$ which is a Nikodým set for $ba(\mathscr {A})$ then there exists $\mathscr {B}_{q}$, with $q\ge p$, such that $\mathscr {B}_{q}$ has (VHS) property. In particular, if $(\mathscr {B}_{m})_{m=1}^{\infty }$ is an increasing covering of a $\sigma$-algebra there exists $\mathscr {B}_{q}$ that has (VHS) property. Valdivia proved that every $\sigma$-algebra has strong Nikodým property and in 2013 asked if Nikodým property in an algebra implies strong Nikodým property. We present three open questions related with this aforementioned Valdivia question and a proof of his strong Nikodým 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.