TITLE: Probability measures on a Polish space
QUESTION [11 upvotes]: Let $X$ be a Polish space, that is a separable metric complete topological space. Is the space of Borel probability measures on $X$, equipped with its weak topology, is Polish too ? 
It is metric, but what about completeness and separability ?

REPLY [9 votes]: Yes, it is (under the topology of weak convergence). This follows from Theorem 6.2 and Theorem  6.5 in Probability Measures on Metric Spaces by K. R. Parthasarathy, which is a good reference for these kind of questions.