Consider two Polish metric probability spaces $(\mathcal{A}, \Sigma_\mathcal{A})$ and $(\mathcal{B}, \Sigma_\mathcal{B})$. 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
$$a\mapsto \kappa(a, \cdot)$$
as a mapping $\mathcal{A}\to\mathcal{P}_\mathcal{B}$. Is this mapping measurable (wrt $\Sigma_{\mathcal{P}_\mathcal{B}}$)?