Skorohod representation theorem via disintegrations

Let $(mu_n:ngeq 0)$ be Borel probabilities on a metric space $S$ such that $mu_n ightarrowmu_0$ weakly. Say that Skorohod representation holds if, on some probability space, there are $S$-valued random variables $X_n$ satisfying $X_nsimmu_n$ for all $n$ and $X_n ightarrow X_0$ in probability. By Skorohod's theorem, Skorohod representation holds (with $X_n ightarrow X_0$ almost uniformly) if $mu_0$ is separable. Two results are proved in this paper. First, Skorohod representation may fail if $mu_0$ is not separable (provided, of course, non separable probabilities exist). Second, independently of $mu_0$ separable or not, Skorohod representation holds if $W(mu_n,mu_0) ightarrow 0$ where $W$ is Wasserstein distance (suitably adapted). The converse is essentially true as well. Such a $W$ is a version of Wasserstein distance which can be defined for any metric space $S$ satisfying a mild condition. To prove the quoted results (and to define $W$), disintegrable probability measures are fundamental.