On a Problem in Measure-Spaces

Let $\mathcal{F}$ be the family of all random variables on a probability space $\Omega$ taking values from a separable and complete metric space $X$. In this paper we prove that $\mathcal{F}$ is in a certain sense a closed family. More precisely, if $\{\xi_n\}$ is a sequence of $X$-valued random variables such that their probability distributions converge weakly to a probability distribution $P$ on $X$, then there exists an $X$-valued random variable on $\Omega$ with distribution $P$. An example is also given which shows that the assumption of completeness of $X$ cannot in general be dropped.