On integration with respect to a DT-measure

Svistula, Marina Genad'evna

doi:10.1007/s11117-015-0373-1

Cited by 9 publications

(8 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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

View full text Add to dashboard Cite

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.

show abstract

Section: New Deficient Topological Measuresmentioning

confidence: 99%

Deficient topological measures on locally compact spaces

Butler

2021

Mathematische Nachrichten

View full text Add to dashboard Cite

show abstract

“…We shall give the proof for the case when X is not compact. (When X is compact the proof is similar but simpler; also, one may use [26,Theorem 4.9].) We shall show that proper simple topological measures are dense in the set of simple measures, and the statements will follow from part (2) of Theorem 37.…”

Section: Density Theoremsmentioning

confidence: 94%

“…When the space is compact, the equivalence of the first two conditions in Theorem 17 and of first three conditions in Theorem 18 was first given in [26,Corollary 4.4,4.5]. When X is compact Theorem 19 was proved in [26], but the method there does not work for a locally compact non-compact space, as the set f −1 ([0, ∞)) = X is not compact. Theorem 20 generalizes results from several papers, including [1], [13], and [26].…”

Section: Sincementioning

confidence: 99%

Weak convergence of topological measures

Butler¹

2019

Preprint

View full text Add to dashboard Cite

Topological measures and deficient topological measures are defined on open and closed subsets of a topological space, generalize regular Borel measures, and correspond to certain non-linear functionals. 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, as well as a version of Prokhorov's Theorem which ensures that every sequence in a family of topological measures has a weakly convergent subsequence. We define Prokhorov's metric and show that convergence in Prokhorov's metric implies weak convergence of a sequence of deficient topological measures. We also generalize many known results about various dense and nowhere dense subsets of deficient topological measures.

show abstract

“…Proof We shall give the proof for the case when X is not compact. (When X is compact, the proof is similar but simpler; also, one may use [41,Theorem 4.9].) We shall show that proper simple topological measures are dense in the set of simple measures, and the statements will follow from part (2) of Theorem 5.1.…”

Section: Theorem 52mentioning

confidence: 95%

Weak Convergence of Topological Measures

Butler

2021

J Theor Probab

View full text Add to dashboard Cite

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.

show abstract

On integration with respect to a DT-measure

Cited by 9 publications

References 9 publications

Deficient topological measures on locally compact spaces

Deficient topological measures on locally compact spaces

Weak convergence of topological measures

Weak Convergence of Topological Measures

Contact Info

Product

Resources

About