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 ?
Probability Theory – Probability Measures on a Polish Space
general-topologyprobabilityprobability theory
Best Answer
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.