Let $(\Omega,\mathcal A),(E,\mathcal E)$ be measurable spaces, $\pi$ be a Markov kernel with source $(E,\mathcal E)$ and target $(\Omega,\mathcal A)$, $(X_t)_{t\ge0}$ be an $(E,\mathcal E)$-valued product measurable process on $(\Omega,\mathcal A)$. Can we conclude that $$E\times[0,\infty)\to\mathbb R\;,\;\;\;(t,x)\mapsto\int\pi(x,\:{\rm d}\omega)f(X_t(\omega))\tag1$$ is $\mathcal E\otimes\mathcal B([0,\infty))$-measurable?
The map in $(1)$ is clearly separately measurable with respect to each argument ...