Skorohod representation on a given probability space

Let $(\Omega,\mathcal{A},P)$ be a probability space, S a metric space, $\mu$ a probability measure on the Borel $\sigma$-field of S, and $X_n:\Omega\rightarrow S$ an arbitrary map, n = 1, 2, . . .. If $\mu$ is tight and $X_n$ converges in distribution 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 $X_n$ are measurable and tight, there are $S$-valued random variables $Z_n$ and $X$, defined on $(\Omega,\mathcal{A},P)$, such that $Z_n\sim X_n$, $X\sim\mu$, and $Z_{n_k}\rightarrow X$ a.s. for some subsequence $(n_k)$. Further, $Z_n\rightarrow X$ a.s. (without need of taking subsequences) if $\mu\{x\}=0$ for all $x$, or if $P(X_n=x)=0$ for some $n$ and all $x$. When $P$ is perfect, the tightness assumption can be weakened into separability up to extending $P$ to $\sigma(\mathcal{A}\cup\{H\})$ for some $H\subset\Omega$ with $P^*(H)=1$. As a consequence, in applying Skorohod representation theorem with separable probability 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 the Borel $\sigma$-field on $(0, 1)$ and $m_H$ the only extension of Lebesgue measure such that $m_H(H)=1$.