Deficient topological measures and functionals generated by them

Mathematische Nachrichten

2021

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.

Section: Introductionmentioning

confidence: 99%

Deficient topological measures on locally compact spaces

Mathematische Nachrichten

2021

“…8]. In particular, there is an order-preserving isomorphism between finite topological measures on X and quasi-linear functionals on C 0 (X ) of finite norm, and μ is a measure iff the corresponding functional is linear (see [15,Theorem 8.7], [37,Theorem 3.9], and [40,Theorem 15]). We outline the correspondence.…”

Section: Remark 13mentioning

confidence: 99%

“…See [ For more examples of topological measures and quasi-integrals on locally compact spaces, see [13,18], and the last section of [19]. For more examples of deficient topological measures, see [17] and [40].…”

Section: Remark 14mentioning

confidence: 99%

“…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%

See 1 more Smart Citation

Weak Convergence of Topological Measures

2021

J Theor Probab

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.

“…(v) The proof is basically one from [18,Theorem 20] and is given here for completeness. Let Z ⊆ X be a zero set.…”

Section: Integrals Over a Set Via Functionalsmentioning

confidence: 99%

Signed Topological Measures on Locally Compact Spaces

2019

Anal Math

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.