On the Skorokhod Representation Theorem

In this paper we present a variant of the well known Skorokhod Representation Theorem. In our main result, given $S$ a Polish space, to a given continous path $\alpha$ in the space of probability measures on $S$, we associate a continuous path in the space of $S$-valued random variables on a nonatomic probability space (endowed with the topology of the convergence in probability). We call this associated path a lifting of $\alpha$. an interesting feature of our result is that we can fix the endpoints ("boundary values") of the lifting of $\alpha$, as long as their distribution correspond to the endpoints ("boundary values") of $\alpha$. We also discuss an $n$-dimensional generalization of this result.