# Probability measure on the set of probability measures

Let $X$ be a metric space, $\mathcal{P}(X)$ be the the set of all Borel probability measures on $X$. I endow $\mathcal{P}(X)$ with the weak* topology.

Question 1: Do I need to impose some further requirement on $X$ for $\mathcal{P}(X)$ to be metrizable?

Let us say that $\mathcal{P}(X)$ (with the weak* topology) is metrizable, and so consider $\mathcal{P}(\mathcal{P}(X))$. Let $\mu \in \mathcal{P}(\mathcal{P}(X))$ and define, for every $B$ borel subset of $X$, $$p_\mu(B)=\int_{\mathcal{P}(X)}\sigma(B)\mu(d\sigma).$$

This is null on the empty set, non-negative and countably additive, so a probability measure on $X$.

Question 2: Am I right or am I missing some nuances?

• If $X$ is separable and metric then $\mathcal{P}(X)$ is also separable and metric Wiki. – Florian Sep 23 '14 at 14:09
• @Florian My question 1, put more explicitly, was whether metrizability of $X$ (with no other requirement) is sufficient for metrizability of $\mathcal{P}(X)$. – fadeaway Sep 23 '14 at 16:46
• I don't know for sure, but I doubt that metrizability of $X$ is sufficient. – Florian Sep 23 '14 at 17:31
• For the definition of $p_{\mu}(B)$ I think you need that the evaluation map $f_B\colon \mathcal{P}(X)\to \mathbb{R}$ defined by $f_B(\sigma)=\sigma(B)$ is $\mu$-measurable -- is that obvious? – Florian Sep 23 '14 at 17:50