On compactness and extreme points of some sets of quasi-measures and measures. II
Miscellaneous properties of the extreme points of the convex set \(S(\mu)\) consisting of all quasi-measure extensions of a quasi-measure \(\mu\), i.e., a non-negative, additive set function on an algebra of subsets of a set to a larger algebra of subsets of the same underlying set, like compactness, denseness in \(S(\mu)\), local compactness, discreteness, zero-dimensionality, connectedness, pathwise connectedness with respect to the weak\(^*\), weak and strong topologies of \(S(\mu)\) are studied. Analogous results are derived for the extreme points of the convex set \(S_\sigma(\mu)\) consisting of all measure extensions of a measure \(\mu\), i.e., a non-negative, \(\sigma\)-additive set function on a \(\sigma\)-algebra of subsets of a set, to a larger \(\sigma\)-algebra of subsets of the same underlying set. In both cases, some results established by the author in Part I of the paper [Manuscr. Math. 86, No. 3, 349-365 (1995; Zbl 0826.28004)] which have the form of an implication are now improved so that they take the form of an equivalence. This is achieved with the help of the antimonogenic component of \(\mu\).
scientific article; zbMATH DE number 871042
scientific article