On Valdivia strong version of Nikodym boundedness property

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• 作者：J. Ka̧kol^a ; ^b ; ^{kakol@amu.edu.pl" class="auth_mail" title="E-mail the corresponding author} ; M. Ló ; pez-Pellicer^c ; ^{mlopezpe@mat.upv.es" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Finitely additive scalar measure ; (LF)(LF)-space ; Nikodym and strong Nikodym property ; Increasing tree ; σ-algebra ; Vector measure
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 February 2017
• 年：2017
• 卷：446
• 期：1
• 页码：1-17
• 全文大小：488 K

文摘

Following Schachermayer, a subset B$\mathcal{B}$ of an algebra A$\mathcal{A}$ of subsets of Ω is said to have the N-property   if a B$\mathcal{B}$-pointwise bounded subset M   of ba(A)$ba(\mathcal{A})$ is uniformly bounded on A$\mathcal{A}$, where ba(A)$ba(\mathcal{A})$ is the Banach space of the real (or complex) finitely additive measures of bounded variation defined on A$\mathcal{A}$. Moreover B$\mathcal{B}$ is said to have the strong N-property   if for each increasing countable covering (B_m)_m${({\mathcal{B}}_m)}_m$ of B$\mathcal{B}$ there exists B_n${\mathcal{B}}_n$ which has the N-property. The classical Nikodym–Grothendieck's theorem says that each σ  -algebra S$\mathcal{S}$ of subsets of Ω has the N-property. The Valdivia's theorem stating that each σ  -algebra S$\mathcal{S}$ has the strong N  -property motivated the main measure-theoretic result of this paper: We show that if (B_m1)_m1${({\mathcal{B}}_{m_1})}_{m_1}$ is an increasing countable covering of a σ  -algebra S$\mathcal{S}$ and if (B_{m1,m2,…,mp,mp+1})_mp+1${({\mathcal{B}}_{m_1,m_2,\dots ,m_p,m_{p+1}})}_{m_{p+1}}$ is an increasing countable covering of B_m1,m2,…,mp${\mathcal{B}}_{m_1,m_2,\dots ,m_p}$, for each p,m_i∈N$p,m_i\in \mathbb{N}$, 1⩽i⩽p$1⩽i⩽p$, then there exists a sequence (n_i)_i${(n_i)}_i$ such that each B_n1,n2,…,nr${\mathcal{B}}_{n_1,n_2,\dots ,n_r}$, r∈N$r\in \mathbb{N}$, has the strong N  -property. In particular, for each increasing countable covering (B_m)_m${({\mathcal{B}}_m)}_m$ of a σ  -algebra S$\mathcal{S}$ there exists B_n${\mathcal{B}}_n$ which has the strong N-property, improving mentioned Valdivia's theorem. Some applications to localization of bounded additive vector measures are provided.