# Weak convergence of topological measures

## TL;DR

This paper extends classical probability results to topological and deficient topological measures, establishing weak convergence criteria, metrics, and generalizations despite their non-linear and non-subadditive nature.

## Contribution

It proves versions of Aleksandrov's and Prokhorov's theorems for these measures, introducing metrics and exploring their convergence properties in topological spaces.

## Key findings

- Weak convergence of deficient topological measures is characterized.
- Prokhorov and Kantorovich-Rubenstein metrics induce weak convergence.
- Many classical probability results extend to non-linear, non-subadditive 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 (non-linear 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 first step to further research in probability theory and its applications in the context of topological measures and corresponding non-linear functionals.