Order-theoretic propertiesof some sets of quasi-measures

Commentationes mathematicae Universitatis Carolinae

Zbigniew Lipecki

关键词: linear lattice;    ideal;    order bounded;    ideal dominated;    order unit;    Banach lattice;    $\\textit{AM}$-space;    convex set;    extreme point;    weakly compact;    additive set function;    quasi-measure;    atomic;    extensionDOI: DOI 10.14712/1213-7243.2015.208AMS Subject Classification: 06F20 28A12 28A33 46A55 46B42 PDF;   

DOI  :  10.14712/1213-7243.2015.208

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 show that $E(\\mu )$ is order bounded if and only if it is contained in a principal ideal in $ba(\\mathfrak R)$ if and only if it is weakly compact and $\\operatorname{extr} E(\\mu )$ is contained in a principal ideal in $ba(\\mathfrak R)$. We also establish some criteria for the coincidence of the ideals, in $ba(\\mathfrak R)$, generated by $E(\\mu )$ and $\\operatorname{extr} E(\\mu )$.

【 授权许可】

CC BY