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On the Skorokhod representation theorem

In this paper we present a variant of the well-known Skorokhod Representation Theorem. First we prove, given S a Polish Space, that to a given continuous path a in the space of probability measures on S, we can 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 a. An interesting feature of our result is that we can fix the endpoints of the lifting of a, as long as their distributions correspond to the respective endpoints of a. Finally, we also discuss and prove an n-dimensional generalization of this result.