Ways of obtaining topological measures on locally compact spaces

S. V. Butler

The Bulletin of Irkutsk State University. Series Mathematics, Tome 25 (2018), pp. 33-45

Résumé

Topological measures and quasi-linear functionals generalize measures and linear functionals. Deficient topological measures, in turn, generalize topological measures. In this paper we continue the study of topological measures on locally compact spaces. For a compact space the existing ways of obtaining topological measures are (a) a method using super-measures, (b) composition of a q-function with a topological measure, and (c) a method using deficient topological measures and single points. These techniques are applicable when a compact space is connected, locally connected, and has a certain topological characteristic, called “genus”, equal to $0$ (intuitively, such spaces have no holes). We generalize known techniques to the situation where the space is locally compact, connected, and locally connected, and whose Alexandroff one-point compactification has genus $0$. We define super-measures and q-functions on locally compact spaces. We then obtain methods for generating new topological measures by using super-measures and also by composing q-functions with deficient topological measures. We also generalize an existing method and provide a new method that utilizes a point and a deficient topological measure on a locally compact space. The methods presented allow one to obtain a large variety of finite and infinite topological measures on spaces such as $ {\mathbb R}^n$, half-spaces in ${\mathbb R}^n$, open balls in ${\mathbb R}^n$, and punctured closed balls in ${\mathbb R}^n$ with the relative topology (where $n \geq 2$).

Détail

Keywords: topological measure, deficient topological measure, solid-set function, super-measure, $q$-function.