Construction and properties of quasi-linear functionals

Johansen, Ørjan; Rustad, Alf Birger

doi:10.1090/s0002-9947-06-03843-8

Cited by 11 publications

(23 citation statements)

References 10 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].…”

Section: New Deficient Topological Measuresmentioning

confidence: 99%

“…In fact, the first paper [6] on the connections between these two fields has been cited over 100 times, and was followed by many articles and chapters in a monograph [12]. Deficient topological measures were first defined and used by A. Rustad and O. Johansen in [10]. They were later independently reintroduced by M. Svistula in [13] (where they were called “functions of class Ψ”) and further developed in [14].…”

Section: Introductionmentioning

confidence: 99%

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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%

Section: Introductionmentioning

confidence: 99%

Deficient topological measures on locally compact spaces

Butler

2021

Mathematische Nachrichten

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“…Deficient topological measures are generalizations of topological measures. They were first defined and used by A. Rustad and O. Johansen [27] and later independently reintroduced and further developed by M. Svistula [39,40]. Deficient topological measures are not only interesting by themselves, but also provide an essential framework for studying topological measures and quasi-linear functionals.…”

Section: Introductionmentioning

confidence: 99%

“…The inclusions follow from the definitions. When X is compact, there are examples of topological measures that are not measures and of deficient topological measures that are not topological measures in numerous papers, beginning with [3,6,12,27], and [39]. When X is locally compact, see [13], [17,Sects.…”

Section: Introductionmentioning

confidence: 99%

Weak Convergence of Topological Measures

Butler

2021

J Theor Probab

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Topological measures and deficient topological measures are defined on open and closed subsets of a topological space, generalize regular Borel measures, and correspond to (nonlinear in general) functionals that are linear on singly generated subalgebras or singly generated cones of functions. They lack subadditivity, and many standard techniques of measure theory and functional analysis do not apply to them. Nevertheless, we show that many classical results of probability theory hold for topological and deficient topological measures. In particular, we prove a version of Aleksandrov’s theorem for equivalent definitions of weak convergence of deficient topological measures. We also prove a version of Prokhorov’s theorem which relates the existence of a weakly convergent subsequence in any sequence in a family of topological measures to the characteristics of being a uniformly bounded in variation and uniformly tight family. We define Prokhorov and Kantorovich–Rubenstein metrics and show that convergence in either of them implies weak convergence of (deficient) topological measures on metric spaces. We also generalize many known results about various dense and nowhere dense subsets of deficient topological measures. The present paper constitutes a necessary step to further research in probability theory and its applications in the context of (deficient) topological measures and corresponding nonlinear functionals.

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“…Дефектные топологические меры (далее DT -меры) являются обобщением топологических мер (далее это T -меры), введенных И. Ф. Аарнесом 1 в [1] для представления квазиинтегралов. Они были выделены независимо в работах 2 [2] и [3] с целью получения ряда результатов для знакопеременных T -мер, поскольку такие инструменты исследования последних, как полная вариация или положительная и отрицательная части, вообще говоря, не являются T -мерами (подробнее см. п.…”

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Дефектные Топологические Меры И Порождаемые Ими Функционалы

Svistula¹

2013

Математический сборник

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Дефектные топологические меры и порождаемые ими функционалы Исследуются свойства дефектных топологических мер, являющихся обобщением топологических мер; изучается интегрирование вещественной непрерывной на компакте функции относительно дефектной топологической меры; введены понятия r-и l-функционалов и получен аналог теоремы Рисса об их представлении. В качестве следствий приведены как известные, так и новые (например, новый вариант определения квазиинтеграла) результаты для квазиинтегралов и топологических мер. Библиография: 16 названий. Ключевые слова: дефектная топологическая мера, топологическая мера, r-и l-функционалы, квазиинтеграл, теорема Рисса о представлении.

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Construction and properties of quasi-linear functionals

Cited by 11 publications

References 10 publications

Deficient topological measures on locally compact spaces

Deficient topological measures on locally compact spaces

Weak Convergence of Topological Measures

Дефектные Топологические Меры И Порождаемые Ими Функционалы

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