Different type of measurability of transition kernel

Question

Let $(E,d)$ be a Polish space equipped with the Borel $\sigma$-algebra $\mathcal{E}$. Let $\mathcal{P}(E)$ be the space of all probability measures on $(E,\mathcal{E})$. We eqiup this space with the topology of weak convergence.

Consider a transition kernel $P(x,\cdot)$, $x\in E$. By definition, for any fixed Borel set $A$, the map $$ x\mapsto P(x,A) $$ is measurable.

My question is whether a more general form of measurability holds. Namely, I am curious whether the map $$ E\ni x\mapsto P(x,\cdot)\in \mathcal{P}(E) $$ is measurable.

Off-top: my motivation for asking this question is as follows. I want to prove that the Wasserstein distance $W_d(P(x,\cdot),P(y,\cdot))$ as a function $E\times E\to\mathbb{R}$ is measurable. To show it, according to Corollary 5.22 of [Villani, "Optimal transport:old and new"], one needs first to establish the property described above.

Answer 1

Yes. The Borel $\sigma$-algebra of the weak topology on $\mathcal{P}(E)$ is generated by the maps $\mu \mapsto \mu(A)$ as $A$ ranges over the Borel sets of $E$. (See for instance Kechris, Classical Descriptive Set Theory, Theorem 17.24.) Since all the maps $x \mapsto P(x,A)$ are measurable by assumption, it follows that the map $x \mapsto P(x, \cdot)$ is measurable as well.