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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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    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](http://books.google.com/books?id=pPv9KCEkklsC&pg=PA112).2012-01-08

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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.