A Skorohod representation theorem without separability

Let (S, d) be a metric space, G a σ-field on S and (μn : n ≥ 0) a sequence of probabilities on G. Suppose G countably generated, the map (x, y) 7→ d(x, y) measurable with respect to G ⊗ G, and μn perfect for n > 0. Say that (μn) has a Skorohod representation if, on some probability space, there are random variables Xn such that Xn ∼ μn for all n ≥ 0 and d(Xn, X0) P −→ 0. It is shown that (μn) has a Skorohod representation if and only if lim n sup f |μn(f)− μ0(f)| = 0, where sup is over those f : S → [−1, 1] which are G-universally measurable and satisfy |f(x) − f(y)| ≤ 1 ∧ d(x, y). An useful consequence is that Skorohod representations are preserved under mixtures. The result applies even if μ0 fails to be d-separable. Some possible applications are given as well.