Weak Convergence of Topological Measures

Abstract

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.

µ-Integrable Functions and Weak Convergence of Probability Measures in Complete Paranormed Spaces

“…Theorem 1 in this article is a modification of Theorem 2.1 and 2.2 in [8]. Theorem 2.1 and 2.2 in [8] require non-negative functions f. Theorem 3 is a generalization of Corollary 1. Corollary 1 is the main result in [1].…”

“…There is a rich bibliography concerning the convergence of sequences of measures, besides the above-mentioned studies [1-3]; see, for example, [4][5][6][7][8].…”

Topological Sigma-Semiring Separation and Ordered Measures in Noetherian Hyperconvexes

“…In other words, the topological determination of measure compactness becomes simpler in this setting. It is shown that topological measures and deficient measures may not always support subadditivity and the properties of linear functionals while admitting the weak convergence of topological measures, which is a variety of Alexandrov weak convergence [9]. Interestingly, if we consider a ring of sets σ(A) and a topological vector space X, then the measure µ : σ(A) → X may show strong convergence to zero if µ( B n i=1 ) → 0 in σ(A) where the sets in sequence B n i=1 under measure are disjoint [10].…”