Mappings and Prohorov spaces

We consider completely regular Hausdorff spaces. In this paper we investigate the space of probability Radon measures P ( X ) on a space X and the property to be a Prohorov space. We prove that the space P ( X ) is sieve-complete if and only if X is sieve-complete. Every mapping φ : X → Y generates the mapping P ( φ ) : P ( X ) → P ( Y ) . Some properties of the mapping P ( φ ) are studied. In particular, we investigate under which conditions the open continuous image of a Prohorov space is Prohorov.