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This article is cited in 5 scientific papers (total in 5 papers)
Ways of obtaining topological measures on locally compact spaces
S. V. Butler University of California Santa Barbara, Santa Barbara, USA
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 $n \geq 2$).
Keywords:
topological measure, deficient topological measure, solid-set function, super-measure, $q$-function.
Received: 10.08.2018
Citation:
S. V. Butler, “Ways of obtaining topological measures on locally compact spaces”, Bulletin of Irkutsk State University. Series Mathematics, 25 (2018), 33–45
Linking options:
https://www.mathnet.ru/eng/iigum344 https://www.mathnet.ru/eng/iigum/v25/p33
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Abstract page: | 133 | Full-text PDF : | 75 | References: | 20 |
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