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 ?

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    $\begingroup$ If probability measures means Borel probability measures then yes. This is proved e.g. in Kechris, Classical descriptive set theory, Theorem 17.23, page 112. $\endgroup$ – t.b. Jan 8 '12 at 16:27

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.


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