Deficient topological measures on locally compact spaces

Butler, Svetlana V.

doi:10.48550/arxiv.1902.02458

Cited by 7 publications

(22 citation statements)

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“…This result follows from Theorem 28 in [5], but it was first observed in an equivalent form for compact-finite topological measures in [14], Proposition 2.2.…”

Section: Preliminariesmentioning

confidence: 57%

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Quasi-linear functionals on locally compact spaces

Butler¹

2019

Preprint

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This paper has two goals: to present some new results that are necessary for further study and applications of quasi-linear functionals, and, by combining known and new results, to serve as a convenient single source for anyone interested in quasi-linear functionals on locally compact non-compact spaces or on compact spaces. We study signed and positive quasi-linear functionals paying close attention to singly generated subalgebras. The paper gives representation theorems for quasi-linear functionals on C c (X) and for bounded quasi-linear functionals on C 0 (X) on a locally compact space, and for quasilinear functionals on C(X) on a compact space. There is an order-preserving bijection between quasi-linear functionals and compact-finite topological measures, which is also "isometric" when topological measures are finite. Finally, we further study properties of quasi-linear functionals and give an explicit example of a quasi-linear functional.

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“…This result follows from Theorem 28 in [5], but it was first observed in an equivalent form for compact-finite topological measures in [14], Proposition 2.2.…”

Section: Preliminariesmentioning

confidence: 57%

“…For more information on proper inclusion, criteria for a topological measure to be a measure from M(X), and examples of finite, compact-finite, and infinite topological measures, see Sections 5 and 6 in [5], Section 9 in [3], and [4].…”

Section: Preliminariesmentioning

confidence: 99%

Quasi-linear functionals on locally compact spaces

Butler¹

2019

Preprint

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“…These include a Representation Theorem showing that quasi-linear functionals are obtained by integration with respect to compact-finite topological measures, continuity of quasi-integrals with respect to the topology of uniform convergence on compacta, and others. These results are obtained in [12], [4], [7], and [6].…”

Section: Introductionmentioning

confidence: 52%

Repeated quasi-integration on locally compact spaces

Butler¹

2019

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When X is locally compact, a quasi-integral (also called a quasi-linear functional) on Cc(X) is a homogeneous, positive functional that is only assumed to be linear on singly-generated subalgebras. We study simple and almost simple quasi-integrals, i.e., quasi-integrals whose corresponding compact-finite topological measures assume exactly two values. We present a criterion for repeated quasi-integration (i.e., iterated integration with respect to topological measures) to yield a quasi-linear functional. We find a criterion for a double quasi-integral to be simple. We describe how a product of topological measures acts on open and compact sets. We show that different orders of integration in repeated quasi-integrals give the same quasi-integral if and only if the corresponding topological measures are both measures or one of the corresponding topological measures is a positive scalar multiple of a point mass.

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“…Let X be a locally compact space whose one-point compactification has genus 0. Let λ be a real-valued topological measure on X (or, more generally, a real-valued deficient topological measure on X; for definition and properties of deficient topological measures on locally compact spaces see [11]). Let P be a set of two distinct points.…”

Section: Examples For a Compact Spacementioning

confidence: 99%

Semisolid sets and topological measures

Butler¹

2021

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This paper is one in a series that investigates topological measures on locally compact spaces. A topological measure is a set function which is finitely additive on the collection of open and compact sets, inner regular on open sets, and outer regular on closed sets. We examine semisolid sets and give a way of constructing topological measures from solid-set functions on locally compact, connected, locally connected spaces. For compact spaces our approach produces a simpler method than the current one. We give examples of finite and infinite topological measures on locally compact spaces and present an easy way to generate topological measures on spaces whose one-point compactification has genus 0. Results of this paper are necessary for various methods for constructing topological measures, give additional properties of topological measures, and provide a tool for determining whether two topological measures or quasi-linear functionals are the same.

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Deficient topological measures on locally compact spaces

Cited by 7 publications

References 0 publications

Quasi-linear functionals on locally compact spaces

Quasi-linear functionals on locally compact spaces

Repeated quasi-integration on locally compact spaces

Semisolid sets and topological measures

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