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Density in the space of topological measures. (English) Zbl 1027.28017
Let \(X\) be a compact Hausdorff space. Denote by \({\mathcal A}(X)\) the family of open or closed subsets of \(X\). A topological measure (formerly “quasi-measure”) on \(X\) is a set function \(\mu : {\mathcal A}(X)\to\mathbb{R}^+\) such that (i) \(\mu(\cup_{i=1}^nA_i)=\sum_{i=1}^n\mu(A_i)\) for any disjoint sets \(A_1,\dots,A_n\in{\mathcal A}(X)\) with \(A=\cup_{i=1}^n A_i\in{\mathcal A}(X)\) and (ii) \(\mu(U)=\sup\{\mu(C) : C\subset U\text{ and }C\text{ closed}\}\) for any open subset \(U\) of \(X\).
There are various types of topological measures such as simple, finitely defined, representable, extreme ones. In this paper the author proves density theorems involving classes of such topological measures which give a way of approximating various topological measures by members of different classes with respect to the weak* topology on the space of topological measures given by quasi-linear functionals.

28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
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