Order boundedness and weak compactnessof the set of quasi-measure extensions of a quasi-measure

Commentationes mathematicae Universitatis Carolinae

Zbigniew Lipecki¹ 

关键词: linear lattice;    order bounded;    additive set function;    quasi-measure;    atomic;    extension;    convex set;    extreme point;    weakly compact;   

DOI  :  10.14712/1213-7243.2015.130

学科分类：物理化学和理论化学

来源: Univerzita Karlova v Praze * Matematicko-Fyzikalni Fakulta / Charles University in Prague, Faculty of Mathematics and Physics

【 摘 要 】

Let $\mathfrak M$ and $\mathfrak R$ be algebras of subsets of a set $\Omega $ with $\mathfrak M\subset\mathfrak R$, and denote by $E(\mu )$ the set of all quasi-measure extensions of a given quasi-measure $\mu $ on $\mathfrak M$ to $\mathfrak R$. We give some criteria for order boundedness of $E(\mu )$ in $ba(\mathfrak R)$, in the general case as well as for atomic $\mu $. Order boundedness implies weak compactness of $E (\mu )$. We show that the converse implication holds under some assumptions on $\mathfrak M$, $\mathfrak R$ and $\mu $ or $\mu $ alone, but not in general.

【 授权许可】

CC BY