Order-theoretic propertiesof some sets of quasi-measures

[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 $\\operatorname{extr} E(\\mu )$.

[学科分类] 物理化学和理论化学

[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