ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2018. Vol. 25

Ways of obtaining topological measures on locally compact spaces

S. V. Butler

Topological measures and quasi-linear functionals generalize measures and linear functionals. Deficient topological measures, in turn, generalize topological mea- sures. 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 com- pact 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 Rn, half-spaces in Rn, open balls in Rn, and punctured closed balls in Rn with the relative topology (where n ≥ 2).

About the Authors
Svetlana V. Butler, Ph. D., Department of Mathematics, 552 University Road, Isla Vista CA 93117, University of California Santa Barbara, Santa Barbara, United States of America, tel.: (805)8932955, e-mail: svetbutler@gmail.com
For citation
Butler S. V. Ways of Obtaining Topological Measures on Locally Compact Spaces. The Bulletin of Irkutsk State University. Series Mathematics, 2018, vol. 25, pp. 33-45. https://doi.org/10.26516/1997-7670.2018.25.33
topological measure, deficient topological measure, solid-set function, super-measure, q-function

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