On Valdivia strong version of Nikodym boundedness property

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:446

Article

Kakol, J.^1,2  Lopez-Pellicer, M.^3,4 

[1] Adam Mickiewicz Univ, Poznan, Poland

[2] Czech Acad Sci, Inst Math, Prague, Czech Republic

[3] Univ Politecn Valencia, Dept Appl Math, Valencia, Spain

[4] Univ Politecn Valencia, IUMPA, Valencia, Spain

关键词: Finitely additive scalar measure;    (LF)-space;    Nikodym and strong Nikodym;    property;    Increasing tree;    sigma-algebra;    Vector measure;   

DOI  :  10.1016/j.jmaa.2016.08.032

来源: Elsevier

【 摘 要 】

Following Schachermayer, a subset B of an algebra A of subsets of Omega is said to have the N-property if a B-pointwise bounded subset M of 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 sigma-algebra S of subsets of Omega has the N-property. The Valdivia's theorem stating that each sigma-algebra S has the strong N-property motivated the main measure-theoretic result of this paper: We show that if (B-m1)(m1) is an increasing countable covering of a sigma-algebra S and if (B-m1,B-m2,B-...,B-mp,B-mp+1)(mp+1) is an increasing countable covering of B-m1,B-m2,B-...,B-mp, for each p, m(i) is an element of N, 1 <= i <= p, then there exists a sequence (n(i))(i) such that each B-n1,B-n2,B-...,B-nr, r is an element of N, has the strong N-property. In particular, for each increasing countable covering (B-m)(m) of a sigma-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. (C) 2016 Elsevier Inc. All rights reserved.

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