Skorohod representation on a given probability space

Let $(\Omega,\mathcal{A},P)$ be a probability space, $S$ a metricspace, $\mu$ a probability measure on the Borel $\sigma$-field of$S$, and $X_n:\Omega\rightarrow S$ an arbitrary map,$n=1,2,\ldots$. If $\mu$ is tight and $X_n$ converges indistribution to $\mu$ (in Hoffmann-J\o rgensen's sense), then$X\sim\mu$ for some $S$-valued random variable $X$ on$(\Omega,\mathcal{A},P)$. If, in addition, the $X_n$ aremeasurable and tight, there are $S$-valued random variables$\overset{\sim}{X}_n$ and $X$, defined on$(\Omega,\mathcal{A},P)$, such that $\overset{\sim}{X}_n\sim X_n$,$X\sim\mu$ and $\overset{\sim}{X}_{n_k}\rightarrow X$ a.s. forsome subsequence $(n_k)$. Further, $\overset{\sim}{X}_n\rightarrowX$ a.s. (without need of taking subsequences) if $\mu\{x\}=0$ forall $x$, or if $P(X_n=x)=0$ for some $n$ and all $x$. When $P$ isperfect, the tightness assumption can be weakened intoseparability up to extending $P$ to $\sigma(\mathcal{A}\cup\{H\})$for some $H\subset\Omega$ with $P^*(H)=1$. As a consequence, inapplying Skorohod representation theorem with separableprobability measures, the Skorohod space can be taken$((0,1),\sigma(\mathcal{U}\cup\{H\}),m_H)$, for some $H\subset(0,1)$ with outer Lebesgue measure 1, where $\mathcal{U}$ is theBorel $\sigma$-field on $(0,1)$ and $m_H$ the only extension ofLebesgue measure such that $m_H(H)=1$. In order to prove theprevious results, it is also shown that, if $X_n$ converges indistribution to a separable limit, then $X_{n_k}$ converges stablyfor some subsequence $(n_k)$.