A Skorohod representation theorem without separability

Let $(S,d)$ be a metric space, $mathcalG$ a $sigma$-field on $S$ and $(mu_n:ngeq 0)$ a sequence of probabilities on $mathcalG$. Suppose $mathcalG$ countably generated, the map $(x,y)mapsto d(x,y)$ measurable with respect to $mathcalGotimesmathcalG$, and $mu_n$ perfect for $n>0$. Say that $(mu_n)$ has a Skorohod representation if, on some probability space, there are random variables $X_n$ such that eginequation* X_nsimmu_n ext for all ngeq 0quad extandquad d(X_n,X_0)oversetPlongrightarrow 0. endequation* It is shown that $(mu_n)$ has a Skorohod representation if and only if eginequation* lim_n,sup_f,absmu_n(f)-mu_0(f)=0, endequation* where $sup$ is over those $f:S ightarrow [-1,1]$ which are $mathcalG$-universally measurable and satisfy $absf(x)-f(y)leq 1wedge d(x,y)$. An useful consequence is that Skorohod representations are preserved under mixtures. The result applies even if $mu_0$ fails to be $d$-separable. Some possible applications are given as well.