Borel complexity of the space of probability measures

Using a technique developed by Louveau and Saint Raymond, we find the complexity of the space of probability measures in the Borel hierarchy: if X X is any non-Polish Borel subspace of a Polish space, then P ( X ) P(X) , the space of probability Borel measures on X X with the weak topology, is always true Π ξ 0 {\boldsymbol {\Pi }^{\boldsymbol {0}}_{\boldsymbol {\xi }}} , where ξ \xi is the least ordinal such that X X is Π ξ 0 {\boldsymbol {\Pi }^{\boldsymbol {0}}_{\boldsymbol {\xi }}} .