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 Σp is a Nikodým set for ba(Σ) was complemented in Ferrando et al. (2020) proving that there exists \(p\in \mathbb {N}\) such that Σp 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.

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Acknowledgements

The second author is supported by Grant PGC2018-094431-B-I00 of the Ministry of Science, Innovation and Universities of Spain.

We thank the reviewers for their useful comments and suggestions.

We also thank to Professor Juan Carlos Ferrando for careful reading the manuscript and valuable discussions and to Professor José Mas for his assistance with LaTex.

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Correspondence to Manuel López-Pellicer.

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To Professor Marco Antonio López at the occasion of his 70th birthday.

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López-Alfonso, S., López-Pellicer, M. Weak Sequential Convergence in Bounded Finitely Additive Measures. Vietnam J. Math. 48, 379–389 (2020). https://doi.org/10.1007/s10013-020-00387-2

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Keywords

  • Algebras and σ-algebras of sets
  • Bounded finitely additive measures
  • Grothendieck
  • Nikodým and Rainwater sets
  • Pointwise and weak sequential convergence
  • Web properties

Mathematics Subject Classification (2010)

  • 28A33
  • 46B25