Quasi-linear functionals on locally compact spaces

Butler, Svetlana V.

doi:10.48550/arxiv.1902.03358

Cited by 4 publications

(15 citation statements)

References 12 publications

(26 reference statements)

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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%

“…There is an order-preserving isomorphism between compact-finite topological measures on X and quasi-integrals on C c (X), and µ is a measure iff the corresponding functional is linear. See Theorem 42 in Section 4 of [7] for this result and Theorem 3.9 in [12] for the first version of the representation theorem. We outline the correspondence.…”

Section: Preliminariesmentioning

confidence: 99%

“…where [a, b] is any interval containing f (X). (See Theorem 30 and Proposition 29 in Section 3 of [7].) (II) Let ρ be a quasi-integral on C c (X).…”

Section: Preliminariesmentioning

confidence: 99%

“…(II) Let ρ be a quasi-integral on C c (X). The corresponding compact-finite topological measure µ = µ ρ is given as follows: For examples of topological measures and quasi-integrals on locally compact spaces see [5] and the last sections of [4] and [7].…”

Section: Preliminariesmentioning

confidence: 99%

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Repeated quasi-integration on locally compact spaces

Butler¹

2019

Preprint

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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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Section: Introductionmentioning

confidence: 52%

Section: Preliminariesmentioning

confidence: 99%

“…where [a, b] is any interval containing f (X). (See Theorem 30 and Proposition 29 in Section 3 of [7].) (II) Let ρ be a quasi-integral on C c (X).…”

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

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Repeated quasi-integration on locally compact spaces

Butler¹

2019

Preprint

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“…The theory of quasi-linear functionals is connected with the theory of Choquet integrals. If µ is a topological measure, the quasi-linear functional X f dµ is a symmetric Choquet integral (see [14], [12] and [15,Ch. 7]).…”

Section: Introductionmentioning

confidence: 99%

Semisolid sets and topological measures

Butler¹

2021

Preprint

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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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Signed Topological Measures on Locally Compact Spaces

Butler

2019

Anal Math

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In this paper we define and study signed deficient topological measures and signed topological measures (which generalize signed measures) on locally compact spaces. We prove that a signed deficient topological measure is τ -smooth on open sets and τ -smooth on compact sets. We show that the family of signed measures that are differences of two Radon measures is properly contained in the family of signed topological measures, which in turn is properly contained in the family of signed deficient topological measures. Extending known results for compact spaces, we prove that a signed topological measure is the difference of its positive and negative variations if at least one variation is finite; we also show that the total variation is the sum of the positive and negative variations. If the space is locally compact, connected, locally connected, and has the Alexandroff one-point compactification of genus 0, a signed topological measure of finite norm can be represented as a difference of two topological measures.2010 Mathematics Subject Classification. Primary 28C15; Secondary 28C99. Key words and phrases. signed deficient topological measure; signed topological measure; positive, negative, total variation of a signed deficient topological measure.

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Quasi-linear functionals on locally compact spaces

Cited by 4 publications

References 12 publications

Repeated quasi-integration on locally compact spaces

Repeated quasi-integration on locally compact spaces

Semisolid sets and topological measures

Signed Topological Measures on Locally Compact Spaces

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