Weak Sequential Convergence in Bounded Finitely Additive Measures

[EN] It is well known that a ¿-algebra ¿ of subsets of a set ¿ verifies both Nikodým property and property (G) for the Banach space ba(¿) of bounded finitely additive measures defined in ¿. A classic result of Valdivia stating that if a ¿-algebra ¿ is covered by an increasing sequence (¿n:n¿N) of subsets, there is p¿N such that ¿p is a Nikodým set for ba(¿) was complemented in Ferrando et al. (2020) proving that there exists p¿N such that ¿p is both a Nikodým and a Grothendieck set for ba(¿). Valdivia result was the first step to get that if (¿¿:¿¿N<¿) is a web in ¿ there exists a chain (¿n:n¿N) in N<¿ such that each ¿¿n, n¿N, is a Nikodým set for ba(¿). 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 (¿¿:¿¿N<¿) in a ¿-algebra ¿ there exists a chain (¿n:n¿N) in N<¿ such that each ¿¿n, n¿N, is both a Nikodým and a Grothendieck set for ba(¿). As an application we extend some results of classic Banach space theoryThe second author is supported by Grant PGC2018-094431-B-I00 of the Ministry of Science, Innovation and Universities of Spain.López Alfonso, S.; López Pellicer, M. (2020). Weak Sequential Convergence in Bounded Finitely Additive Measures. Vietnam Journal of Mathematics. 48(2):379-389. https://doi.org/10.1007/s10013-020-00387-2379389482Arens, R.F., Kelley, J.L.: Characterizations of the space of continuous functions over a compact Hausdorff space. Trans. Am. Math. Soc. 62, 499–508 (1947)Diestel, J., Faires, B., Huff, R.: Convergence and boundedness of measures in non σ-complete algebras. Preprint (1976)Diestel, J., Uhl, J.J.: Vector Measures. Mathematical Surveys and Monographs, vol. 15. American Mathematical Society, Providence (1977)Dunford, N., Schwartz, J.T.: Linear Operators. Part I: General Theory. Wiley, New Jersey (1988)Fernández, J., Hui, S., Shapiro, H.: Unimodular functions and uniform boundedness. Publ. Mat. 33, 139–146 (1989)Ferrando, J.C., Ka̧kol, J., López-Pellicer, M.: On spaces Cb(X) weakly K-analytic. Math. Nachr. 290, 2612–2618 (2017)Ferrando, J.C., López-Alfonso, S., López-Pellicer, M.: On Nikodým and Rainwater sets for $ba(\mathcal {R})$ and a problem of M. Valdivia. Filomat 33, 2409–2416 (2019)Ferrando, J.C., López-Alfonso, S., López-Pellicer, M.: On the Grothendieck property (submited) (2020)Ferrando, J.C., López-Pellicer, M., Sánchez Ruiz, L.M.: Metrizable Barrelled Spaces. Pitman Research Notes in Mathematics Series, vol. 332. Longman, Harlow (1995)Ferrando, J.C., Sánchez Ruiz, L.M.: A survey on recent advances on the Nikodým boundedness theorem and spaces of simple functions. Rocky Mount. J. Math. 34, 139–172 (2004)Fonf, V.P.: On exposed and smooth points of convex bodies in Banach spaces. Bull. Lond. Math. Soc. 28, 51–58 (1996)Ka̧kol, J., López-Pellicer, M.: On Valdivia strong version of Nikodým boundedness property. J. Math. Anal. Appl. 446, 1–17 (2017)López-Alfonso, S., Mas, J., Moll, S.: Nikodým boundedness property and webs in σ-algebras. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 110, 711–722 (2016)López-Alfonso, S.: On Schachermayer and Valdivia results in algebras of Jordan measurable sets. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 110, 799–808 (2016)López-Pellicer, M.: Webs and bounded finitely additive measures. J. Math. Anal. Appl. 210, 257–267 (1997)Nygaard, O.: A strong uniform boundedness principle in Banach spaces. Proc. Am. Math. Soc. 129, 861–863 (2001)Nygaard, O.: Thick sets in Banach spaces and their properties. Quaest. Math. 29, 50–72 (2006)Plebanek, G., Sobota, D.: Countable tightness in the spaces of regular probability measures. Fund. Math. 229, 159–170 (2015)Rainwater, J.: Short notes: Weak convergence of bounded sequences. Proc. Am. Math. Soc. 14, 999–999 (1963)Schachermayer, W.: On some classical measure-theoretic theorems for non-sigma-complete Boolean algebras. Diss. Math. (Rozprawy Mat.) 214, 1–33 (1982)Simons, S.: A convergence theorem with boundary. Pac. J. Math. 40, 703–708 (1972)Sobota, D., Zdomskyy, L.: The Nikodým property in the Sacks model. Topol. Appl. 230, 24–34 (2017)Talagrand, M.: Propriété de Nikodým and propriété de Grothendieck. Stud. Math. 78, 165–171 (1984)Valdivia, M.: On certain barrelled normed spaces. Ann. Inst. Fourier 29, 39–56 (1979)Valdivia, M.: On Nikodým boundedness property. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 107, 355–372 (2013