Nikodym boundedness property for webs in \(\sigma \) -algebras
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- 作者：S. López-Alfonso ; J. Mas ; S. Moll
- 刊名：Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas
- 出版年：2016
- 出版时间：September 2016
- 年：2016
- 卷：110
- 期：2
- 页码：711-722
- 全文大小：481 KB
- 刊物类别：Mathematics and Statistics
- 出版者：Springer Milan
- ISSN：1579-1505
- 卷排序：110
文摘
A subset \(\mathscr {B}\) of an algebra \(\mathscr {A}\) of subsets of \(\Omega \) is said to have property N if a \(\mathscr {B}\)-pointwise bounded subset M of \(\mathrm{ba}(\mathscr {A})\) is uniformly bounded on \(\mathscr {A}\), where \(\mathrm{ba}(\mathscr {A})\) is the Banach space of the real (or complex) finitely additive measures of bounded variation defined on \(\mathscr {A}\) with the norm variation. Moreover \(\mathscr {B}\) is said to have property sN if for each increasing countable covering \((\mathscr {B}_{m})_{m}\) of \(\mathscr {B}\) there exists \(\mathscr {B}_{n}\) which has property N and \(\mathscr {B}\) is said to have property wN if given the increasing countable coverings \((\mathscr {B}_{m_{1}})_{m_{1}}\) of \(\mathscr {B}\) and \((\mathscr {B}_{m_{1}m_{2}\ldots m_{p}m_{p+1}})_{m_{p+1}}\) of \(\mathscr {B}_{m_{1}m_{2}\ldots m_{p}}\), for each \(p,m_{i}\in \mathbb {N}\), \(1 \leqslant i \leqslant p+1\), there exists a sequence \((n_{i})_{i}\) such that each \(\mathscr {B}_{n_{1}n_{2}\ldots n_{r}}\), \(r\in \mathbb {N}\), has property N. For a \(\sigma \)-algebra \(\mathscr {S}\) of subsets of \(\Omega \) it has been proved that \(\mathscr {S}\) has property N (Nikodym–Grothendieck), property sN (Valdivia) and property w(sN) (Kakol–López-Pellicer). We give a proof of property wN for a \(\sigma \)-algebra \(\mathscr {S}\) which is independent of properties N and sN. This result and the equivalence of properties wN and \(w^{2}N\) enable us to give some applications to localization of bounded additive vector measures.KeywordsBounded setfinitely additive scalar (vector) measureinductive limitNV-tree\(\sigma \)-algebraweb Nikodym property