Abstract:
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 $n \geq 2$).
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