Convergence and Existence of Random Set Distributions

We study the relation between distributions of random closed sets and their hitting functions $T$, defined by $T(B) = P\{\varphi \cap B \neq \varnothing\}$ for Borel sets $B$. In particular, a sequence of random sets converges in distribution iff the corresponding sequence of hitting functions converges on some sufficiently large class of bounded Borel sets. This class may be chosen to be countable.