Density in the space of topological measures

Butler, Svetlana V.

doi:10.4064/fm174-3-4

Cited by 9 publications

(6 citation statements)

References 9 publications

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“…When X is compact the proper inclusion in (2) was first demonstrated in [1]; in fact, M(X) is nowhere dense in T M(X) (see [2]).…”

Section: Preliminariesmentioning

confidence: 99%

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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“…When X is compact the proper inclusion in (2) was first demonstrated in [1]; in fact, M(X) is nowhere dense in T M(X) (see [2]).…”

Section: Preliminariesmentioning

confidence: 99%

Quasi-linear functionals on locally compact spaces

Butler¹

2019

Preprint

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“…There exists a nonzero finite proper deficient topological measure that is not a topological measure. [11] from a compact space to a locally compact one.…”

Section: Lemma 51 Suppose X Is Locally Compact ∞mentioning

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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“…15], and [13]. When X is compact the proper inclusion in (1.1) was first demonstrated in [3]; in fact, M (X) is nowhere dense in T M (X) (see [10]).…”

Section: (Tm1) Ifmentioning

confidence: 99%

“…(A quasi-linear functional corresponding to a simple topological measure is multiplicative on singly generated subalgebras, see [4], [20]). From the point of view of the theory of topological measures, the explanation of the "non-approximation" results from [22] is suggested by [10,Cor. 4.13] or by the fact that approximation of a quasi-state, in general, requires a much larger collection than just quasi-integrals corresponding to simple topological measures.…”

Section: Properties Of Quasi-integralsmentioning

confidence: 99%

Quasi-linear functionals on locally compact spaces

Butler¹

2021

Confluentes Mathematici

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Quasi-linear functionals on locally compact spaces Tome 13, n o 1 (2021), p. 3-34. © Les auteurs et Confluentes Mathematici, 2021. Certains droits réservés. Cet article est mis à disposition selon les termes de la Licence Internationale d'Attibution Creative Commons BY-NC-ND 4.0 https://creativecommons.org/licenses/by/4.0 L'accès aux articles de la revue « Confluentes Mathematici » (http://cml.centre-mersenne.org/) implique l'accord avec les conditions générales d'utilisation (http://cml.centre-mersenne. org/legal/). Confluentes Mathematici est membre du Centre Mersenne pour l'édition scientifique ouverte http:// www.centre-mersenne.org/

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Density in the space of topological measures

Cited by 9 publications

References 9 publications

Quasi-linear functionals on locally compact spaces

Quasi-linear functionals on locally compact spaces

Weak Convergence of Topological Measures

Quasi-linear functionals on locally compact spaces

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