A survey on Skorokhod representation theorem without separability

Let $S$ be a metric space, $mathcalG$ a $sigma$-field of subsets of $S$ and $(mu_n:ngeq 0)$ a sequence of probability measures on $mathcalG$. Say that $(mu_n)$ admits a Skorokhod representation if, on some probability space, there are random variables $X_n$ with values in $(S,mathcalG)$ such that eginequation* X_nsimmu_n ext for each nge 0quad extandquad X_n ightarrow X_0 ext in probability. endequation* We focus on results of the following type: $(mu_n)$ has a Skorokhod representation if and only if $J(mu_n,mu_0) ightarrow 0$, where $J$ is a suitable distance (or discrepancy index) between probabilities on $mathcalG$. One advantage of such results is that, unlike the usual Skorokhod representation theorem, they apply even if the limit law $mu_0$ is not separable. The index $J$ is taken to be the bounded Lipschitz metric and the Wasserstein distance.