Weak Sequential Convergence in Bounded Finitely Additive Measures

Abstract

It is well known that a σ-algebra Σ of subsets of a set Ω verifies both Nikodým property and property (G) for the Banach space ba(Σ) of bounded finitely additive measures defined in Σ. A classic result of Valdivia stating that if a σ-algebra Σ is covered by an increasing sequence $({\Sigma }_{n}:n\in \mathbb {N})$ of subsets, there is $p\in \mathbb {N}$ such that Σ_p is a Nikodým set for ba(Σ) was complemented in Ferrando et al. (2020) proving that there exists $p\in \mathbb {N}$ such that Σ_p is both a Nikodým and a Grothendieck set for ba(Σ). Valdivia result was the first step to get that if $({\Sigma }_{\sigma }:\sigma \in \mathbb {N}^{<\infty })$ is a web in Σ there exists a chain $(\sigma _{n}:n\in \mathbb {N})$ in $\mathbb {N}^{<\infty }$ such that each ${\Sigma }_{\sigma _{n}}$, $n\in \mathbb {N}$, is a Nikodým set for ba(Σ). In this paper, we develop several properties in Banach spaces that enables us to complement the preceding web result extending the main result in Ferrando et al. (2020) proving that for each web $({\Sigma }_{\sigma }:\sigma \in \mathbb {N}^{<\infty })$ in a σ-algebra Σ there exists a chain $(\sigma _{n}:n\in \mathbb {N})$ in $\mathbb {N}^{<\infty }$ such that each ${\Sigma }_{\sigma _{n}}$, $n\in \mathbb {N}$, is both a Nikodým and a Grothendieck set for ba(Σ). As an application we extend some results of classic Banach space theory.