Abstract: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 $\operatorname{extr} E(\mu )$.
Keywords: 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; extension
DOI: DOI 10.14712/1213-7243.2015.208
AMS Subject Classification: 06F20 28A12 28A33 46A55 46B42