zbMATH — the first resource for mathematics

Ways of obtaining topological measures on locally compact spaces. (English) Zbl 1409.28005
Summary: 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 \(n \geq 2\)).

28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
22D05 General properties and structure of locally compact groups
Full Text: DOI Link
[1] Aarnes, J. F., Quasi-states and quasi-measures, Adv. Math. , 1, 41-67, (1991) · Zbl 0744.46052
[2] Aarnes, J. F., Pure quasi-states and extremal quasi-measures, Math. Ann. , 575-588, (1993) · Zbl 0791.46028
[3] Aarnes, J. F., Construction of non-subadditive measures and discretization of Borel measures, Fundamenta Mathematicae , 213-237, (1995) · Zbl 0842.28004
[4] Aarnes, J. F.; Butler, S. V., Super-measures and finitely defined topological measures, Acta Math. Hungar. , 33-42, (2003) · Zbl 1026.28014
[5] Aarnes, J. F.; Rustad, A. B., Probability and quasi-measures - a new interpretation, Math. Scand , 2, 278-284, (1999) · Zbl 0967.28014
[6] Butler, S. V., Q-functions and extreme topological measures, J, Math. Anal. Appl. , 465-479, (2005) · Zbl 1074.28007
[7] Butler, S. V., Solid-set functions and topological measures on locally compact spaces, submitted
[8] Butler, S. V., Deficient topological measures on locally compact spaces, submitted
[9] Grubb, D. J., Irreducible partitions and the construction of quasi-measures, Trans. Amer. Math. Soc. , 5, 2059-2072, (2001) · Zbl 0968.28007
[10] Johansen, Ø.; Rustad, A., Construction and properties of quasi-linear func- tionals, Trans. Amer. Math. Soc. , 6, 2735-2758, (2006) · Zbl 1111.28013
[11] Knudsen, F. F., Topology and the construction of extreme quasi-measures, Adv. Math. , 2, 302-321, (1996) · Zbl 0914.28010
[12] Entov, M.; Polterovich, L., Quasi-states and symplectic intersections, ArXiv , (2004)
[13] Polterovich, L.; Rosen, D., Function theory on symplectic manifolds. CRM Monograph series , (2014)
[14] Svistula, M. G., A signed quasi-measure decomposition, Vestnik SamGU. Estestvennonauchnaia seriia , 3, 192-207, (2008) · Zbl 1321.28026
[15] Svistula, M. G., Deficient topological measures and functionals generated by them, Sbornik: Mathematics , 5, 726-761, (2013) · Zbl 1279.28018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.