# zbMATH — the first resource for mathematics

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.)
Full Text: