Quasi-measures with finitely or countably many extreme extensions
Let \(E(\mu)\) denote the convex set consisting of all quasi-measure extensions of a quasi-measure \(\mu\), i.e., a positive additive function on an algebra \({\mathcal M}\) of subsets of a set \(\Omega\), to a larger algebra \({\mathcal R}\) of subsets of \(\Omega\). It is shown that the set \(\text{extr }E(\mu)\) consisting of the extreme points of \(E(\mu)\) is finite if and only if \(E(\mu)\) is finite-dimensional and that \(E(\mu)\) is finite-dimensional if and only if \(E(\mu)\) is affinely isomorphic to a finite product of finite-dimensional simplices. Moreover, it is proved in the case where \({\mathcal M}\) and \({\mathcal R}\) are \(\sigma\)-algebras and \(\mu\) is a measure that \(\text{extr }E(\mu)\) is either finite or has cardinality not smaller than the cardinality of \(\mathbb{R}\). In particular, \(E(\mu)\) consists only of measures in the case where \(\text{extr }E(\mu)\) is finite and the property of \(\mu\) that \(E(\mu)\) consists only of measures together with the property of \(\mu\) to have finite range implies that \(\text{extr }E(\mu)\) is finite.
scientific article; zbMATH DE number 1239591