For a measurable space $(\Omega_i,\mathcal F_i)$, $i=1,2,3$, consider a stochastic kernel $K<\infty$ on a measurable space on $(\Omega_2,\mathcal F_3)$, and consider $g$ which is an $\mathcal F_1 \times \mathcal F_3$ measurable function. Then we want to prove that if the following integral exists: $$ h(\omega_1,\omega_2) = \int_{\Omega_3} g(\omega_1,\omega_3)K(\omega_2,d\omega_3)$$ then for each $\omega_i \in \Omega_i$ such that $i=1,2$, then $h$ is $\mathcal{F}_1\times \mathcal{F}_2$ measurable.
- I think that we need to show that for every $E \in \mathcal F_2$, then $h^{-1}(E) \in \mathcal F_1$? If so, I think $h^{-1} \in \Omega_1$ as $g\in \Omega_1?$
- I know that for a Kernel defined on on $(\Omega_2,\mathcal F_3)$, then the kernel is a measure on $\mathcal{F}_3$ I think? So if $h$ is an integral of the product of a $\mathcal F_3$ and a $\mathcal F_1 \ \mathcal F_3$ measurable function, can we do something here? I don't really know how to proceed...
You may use a version of the FMCT (Theorem 5.2.2 here). Let $\mathcal{H}$ be the set of functions $g$ for which $h$ is $\mathcal{F}_1\otimes \mathcal{F}_2$-measurable. (1) For $g(\omega_1,\omega_3)=1_A(\omega_1)1_B(\omega_3)$, where $A\in \mathcal{F}_1$ and $B\in\mathcal{F}_3$, $$ h(\omega_1,\omega_2)=1_A(\omega_1)\int_B K(\omega_2,d\omega_3)=1_A(\omega_1)K(\omega_2,B) $$ is clearly $\mathcal{F}_1\otimes \mathcal{F}_2$-measurable and so $1_{A\times B}\in \mathcal{H}$. (2) $\mathcal{H}$ is stable under addition and pointwise increasing limits (by the linearity of $\int$ and the monotone convergence theorem). Therefore, $\mathcal{H}$ contains all bounded measurable functions (for an unbounded $g$ use $(g \wedge M)\vee (-M)$ and send $M\to\infty$).