  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.
