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 ?
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 ?
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.