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

Lipecki, Zbigniew

Commentationes Mathematicae Universitatis Carolinae, Tome 56 (2015) no. 3, pp. 331-345.

Résumé

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.

Détail

DOI : 10.14712/1213-7243.2015.130

Classification : 06F20, 28A12, 28A33, 46A55, 46B42
Keywords: linear lattice; order bounded; additive set function; quasi-measure; atomic; extension; convex set; extreme point; weakly compact