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Signed topological measures on locally compact spaces. (English) Zbl 1449.28014
In this paper a (signed) topological measure on a locally compact space \(X\) is a function \(\mu\) defined on the union of the families of open sets, \(\mathcal{O}(X)\), and closed sets, \(\mathcal{C}(X)\), with values in \([0,\infty]\) (in \([-\infty,\infty]\)) that is finitely additive on \(\mathcal{O}(X)\cup\mathcal{K}(X)\), where \(\mathcal{K}(X)\) is the family of compact sets. It is also required to satisfy two regularity conditions: if \(U\) is open then \(\mu(U)=\lim\{\mu(K):K\in\mathcal{K}(X), K\subseteq U\}\) and if \(F\) is closed then \(\mu(F)=\lim\{\mu(O):O\in\mathcal{O}(X), F\subseteq O\}\), where the limit is taken along the family on the right hand side, directed by (reverse) inclusion.
If \(\mu\) is only required to be additive on \(\mathcal{K}(X)\) then it is called a (signed) deficient topological measure.
The author proves some structural results on these measures: they are the difference of their positive and negative variations; the latter add up to the total variation. In special cases a signed topological measure can be written as the difference of two topological easures: if \(X\) is connected, locally connected and its one-point compactification has genus \(0\).
Reviewer: K. P. Hart (Delft)

MSC:
28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
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