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 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.

Semisolid sets and topological measures

“…It is also possible to define Prokhorov and Kantorovich-Rubenstein metrics and show that convergence in either of them implies weak convergence of topological measures. See [13].…”