A strong version of the Skorohod representation theorem

For each $nge 0$, let $mu_n$ be a tight probability measure on the Borel $sigma$-field of a metric space $S$. Let $(T,mathcal{C})$ be a measurable space such that the diagonal $igl{(t,t):tin Tigr}$ belongs to $mathcal{C}otimesmathcal{C}$. Fix a measurable function $g:S ightarrow T$ and suppose $mu_n=mu_0$ on $g^{-1}(mathcal{C})$ for all $nge 0$. Necessary and sufficient conditions for the existence of $S$-valued random variables $X_n$, defined on the same probability space and satisfying egin{gather*} X_noverset{a.s.}longrightarrow X_0,quad X_nsimmu_n, ext{ and },g(X_n)=g(X_0), ext{ for all }nge 0, end{gather*} are given. Such conditions are then applied to several examples. The tightness condition on $mu_0$ can be dropped at the price of some assumptions on $S$ and $g$.