On Valdivia strong version of Nikodym boundedness property

Abstract

[EN] Following Schachermayer, a subset B of an algebra A of subsets of Ω is said to have the N-property if a B-pointwise bounded subset Mof ba(A)is uniformly bounded on A, where ba(A) is the Banach space of the real (or complex) finitely additive measures of bounded variation defined on A. Moreover B is said to have the strong N-property if for each increasing countable covering (B_m)_m of B there exists B_n which has the N-property. The classical Nikodym Grothendieck s theorem says that each σ-algebra S of subsets of Ω has the N-property. The Valdivia s theorem stating that each σ-algebra S has the strong N-property motivated the main measure-theoretic result of this paper: We show that if (B_{m_1})_{m_1} is an increasing countable covering of a σ-algebra S and if (B_{m_1},_{m_2},...,_{m_p}_{m_(p+1)}}_{m_(p+1)} is an increasing countable covering of B_{m_1},_{m_2},...,_{m_p}, for each p, m_i \in N, 1 less than or equal i less than or equal p, then there exists a sequence (n_i)_i such that each B_{n_1},_{n_2},...,_{n_r}, r∈N, has the strong N-property. In particular, for each increasing countable covering (B_m)_m of a σ-algebra S there exists B_n which has the strong N-property, improving mentioned Valdivia s theorem. Some applications to localization of bounded additive vector measures are provided.