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 $X$ is any non-Polish Borel subspace of a Polish space, then $P(X)$, the space of probability Borel measures on $X$ with the weak topology, is always true ${\boldsymbol{\Pi}^{\boldsymbol{0}}_{\boldsymbol{\xi}}}$, where $\xi$ is the least ordinal such that $X$ is ${\boldsymbol{\Pi}^{\boldsymbol{0}}_{\boldsymbol{\xi}}}$.