Supports of quasi-measures (Q2705792)

In this paper, a quasi-measure on a compact Hausdorff space \(X\) means a real-valued, nonnegative set function \(\mu\) defined on \({\mathcal F}\) (the collection of all closed subsets of \(X\)) that satisfies the following three conditions:NEWLINENEWLINENEWLINE(1) If \(F,F'\in{\mathcal F}\), \(F\subset F'\), then \(\mu(F)\leq \mu(F')\).NEWLINENEWLINENEWLINE(2) If \(F, F'\in{\mathcal F}\), \(F\cap F'= \emptyset\), then \(\mu(F\cup F')= \mu(F)+ \mu(F')\).NEWLINENEWLINENEWLINE(3) If \(F\in{\mathcal F}\) and \(\varepsilon> 0\), then there exists an \(F'\in{\mathcal F}\) such that \(F'\cap F= \emptyset\) and \(\mu(F)+ \mu(F')> \mu(X)- \varepsilon\).NEWLINENEWLINENEWLINEThen, the author introduces a notion of support for such quasi-measures that generalizes the usual notion of support for Borel measures and investigates the relationship between supports of quasi-measures and connectivity of the underlying space \(X\). He also proves a theorem on decompositions of quasi-measures that implies, for example, that if there exists a fully supported simple quasi-measure on \(X\), then \(X\) must be connected and (if also locally connected) no closed \(0\)-dimensional set disconnects \(X\).