Open Mappings of Probability Measures and the Skorokhod Representation Theorem

We prove that for the wide class of spaces X and Y (including completely regular Souslin spaces), every open surjective mapping $f\colon X\to Y$ induces the open mapping $\hat f\colon\mu\mapsto\mu\circ f^{-1}$ between the spaces of probability measures ${\mathcal P} (X)$ and ${\mathcal P} (Y)$. We discuss the existence of continuous inverse mappings for $\hat f$ and connections with the Skorokhod representation theorem and its generalizations.