Consider two Polish metric probability spaces $(\mathcal{A}, \Sigma_\mathcal{A})$ and $(\mathcal{B}, \Sigma_\mathcal{B})$, endowed with their Borel $\sigma$-algebras. Denote as $\mathcal{P}_\mathcal{B}$ the space of probability measures on $(\mathcal{B}, \Sigma_\mathcal{B})$. We can endow it with the Borel $\sigma$ algebra $\Sigma_{\mathcal{P}_\mathcal{B}}$ generated by the weak topology (wrt the weak convergence of measures). 

Consider a Markov kernel $\kappa:\mathcal{A}\times\Sigma_\mathcal{B}\to [0,1]$. In particular we can see
$$a\mapsto \mu_a=\kappa(a, \cdot)$$
as a mapping $\mathcal{A}\to\mathcal{P}_\mathcal{B}$. Is this mapping measurable (wrt $\Sigma_{\mathcal{P}_\mathcal{B}}$)?

The main motivation behind this question is that it is related to the following problem (cf. https://math.stackexchange.com/questions/4360086). Denoted as $\mathcal W$ the 1-Wasserstein distance between probabilities in $\mathcal P_\mathcal B$, defined via the metric on $\mathcal B$, fixed $\nu\in\mathcal P_\mathcal B$, is the mapping $\mathcal A\to\mathbb R$ 
$$a\mapsto \mathcal W(\nu, \mu_a)$$
always measurable?

I think that one way to prove the measurability of this last function would be to exploit Corollary 5.22 in [1], which essentially tells you that if $a\mapsto\mu_a$ is measurable wrt $\Sigma_{\mathcal{P}_\mathcal{B}}$, then $a\mapsto\pi_a$ is measurable, where $\pi_a$ is the optimal coupling between $\mu_a$ and $\nu$. It would then follow that $a\mapsto \mathcal{W}(\mu_a, \nu) = \mathbb E_{(A,A')\sim\pi_a}[d(A, A')]$ is measurable. So we are back to the first question: is it actually true that $a\mapsto\mu_a$ is measurable?

I am interested in this since I have often encountered expressions like 
$$\int_\mathcal A \mathcal{W}(\mathbb P_B, \mathbb P_{B|A=a})\,\mathrm d\mathbb P_A(a)$$
(where $A, B$ are coupled random variables, with marginals $\mathbb P_A$ and $\mathbb P_B$, and $\mathbb P_{B|A=a}$ is a regular conditional probability) without any formal justification, see for instance [2] and the papers it builds on. But does this expression actually make sense? Is the integrand always measurable?


[1] Villani, Optimal transport, old and new, 2008.  
[2] Rodríguez-Gálvez, Tighter expected generalization error bounds via Wasserstein Distance, 2021.