Let $(X,\mathcal{F})$ and $(Y,\mathcal{G})$ be two measurable space. A transition kernel $K$ is a function $K : X \times \mathcal{G} \to \overline{\mathbb{R}}_+$ suche that $K(\cdot,B)$ is measurable and $K(x,\cdot)$ is a measure for every $B \in \mathcal{G}$ and $x \in X$. We said that is a markov kernel if the measures $K(x,\cdot)$ are probability measures.
I was having problems to prove that the function $$x\to K(x,C^1(x))$$ is measurable for every $C \in \mathcal{F} \otimes \mathcal{G}$ in the general case. When we have a markov kernel is easy to prove this using $\pi-\lambda$ theorem but in the not finite case I don't know if is even true.
Any help or reference will be appriciated
Coment: If we prove that $x\to K(x,C^1(x))$ is measurable and we have a measure $\mu$ in $X$ we can define the measure $$\nu(C)=\int_X K(x,C^1(x))d\mu(x)$$ and it will be the unique measure in $\mathcal{F} \otimes \mathcal{G}$ such that $$\nu(A\times B)=\int_A K(x,B)d\mu(x)$$