Additivity of quasi-measures

Grubb, D. J.; LaBerge, Tim

doi:10.1090/s0002-9939-98-04494-3

Cited by 18 publications

(4 citation statements)

References 6 publications

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“…Remark Proposition 5.1 was first proved for both spaces compact and ν finite in [10, Proposition 27]. When X is compact, topological measures (deficient topological measures) that are restrictions of topological measures (deficient topological measures) to sets appeared in several papers, including [7], [9], [14], [15]. Proposition 5.3, Theorem 5.7, Lemma 5.8, and Theorem 5.9 are generalizations to a locally compact case of [14, Proposition 3], [15, Proposition 5.1], and the stated without proof part (4) of [15, Proposition 5.2].…”

Section: New Deficient Topological Measuresmentioning

confidence: 99%

Deficient topological measures on locally compact spaces

Butler

2021

Mathematische Nachrichten

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Topological measures and quasi‐linear functionals generalize measures and linear functionals. We define and study deficient topological measures on locally compact spaces. A deficient topological measure on a locally compact space is a set function on open and closed sets which is finitely additive on compact sets, inner regular on open sets, and outer regular on closed sets. Deficient topological measures generalize measures and topological measures. First we investigate positive, negative, and total variation of a signed set function that is only assumed to be finitely additive on compact sets. These positive, negative, and total variations turn out to be deficient topological measures. Then we examine finite additivity, superadditivity, smoothness, and other properties of deficient topological measures. We obtain methods for generating new deficient topological measures. We provide necessary and sufficient conditions for a deficient topological measure to be a topological measure and to be a measure. The results presented are necessary for further study of topological measures, deficient topological measures, and corresponding non‐linear functionals on locally compact spaces.

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Section: New Deficient Topological Measuresmentioning

confidence: 99%

Deficient topological measures on locally compact spaces

Butler

2021

Mathematische Nachrichten

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“…Remark 45. When X is compact, topological measures (deficient topological measures) that are restrictions of topological measures (deficient topological measures) to sets appeared in several papers, including [7], [9], [13], [14]. Proposition 36, Theorem 42, and Lemma 43 are generalizations to a locally compact case of Proposition 3 in [13],…”

Section: New Deficient Topological Measuresmentioning

confidence: 99%

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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“…В связи с этим полуаддитивную квазимеру принято называть мерой. Примеры квазимер, не являющихся мерами, содержатся в работах [1], [3], [5], [6] и других.…”

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“…2 мы рассматриваем функцию ϕ µ , порождаемую квазимерой µ по аналогии с внешней мерой, индуцированной мерой; с помощью этой функции в п. 3 передоказываем теорему о разложении квазимеры, при этом, в отличие от известного доказательства [5], получаем выражение слагаемого -меры через квазимеру; в п. 4, опираясь на теорему о разложении, даем критерий правильной квазимеры; с помощью этого критерия в п.…”

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Критерий Правильной Квазимеры

Svistula¹

2007

Mat. Zametki

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В известной теореме о разложении квазимеры в сумму меры и правильной квазимеры дано новое представление слагаемого-меры, что позволило получить критерий правильной квазимеры. С помощью этого критерия решена проблема о сумме правильных квазимер. Полученные результаты перенесены на квазисостояния. Библиография: 6 названий.

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Additivity of quasi-measures

Cited by 18 publications

References 6 publications

Deficient topological measures on locally compact spaces

Deficient topological measures on locally compact spaces

Signed Topological Measures on Locally Compact Spaces

Критерий Правильной Квазимеры

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