Let $P:\mathbb{R}^n\times \mathscr{B}_{\mathbb{R}^n} \rightarrow [0,\infty]$ be a function such that $P(x,-)$ is a probability measure for each $x$ and $P(-,A)$ is (borel) measurable for each $A$.
Let $A\in \mathscr{B}_{\mathbb{R}^n}\otimes \mathscr{B}_{\mathbb{R}^n}$. Then, how do I prove that the map $x\mapsto P(x,A_x)$ is measurable? ($A_x$ denotes the $x$-section of $A$)
Let $A$ and $B$ be Borel sets. Then $P(x,(A\times B)_x)=P(x,B)$ if $x \in A$ and $0$ if $x \notin A$, so $P(x,(A\times B)_x)$ is measurable. If $E$ is a finite disjoint union of measurable rectangles $A\times B$ then $P(x,E_x)$ is a finite sum of functions of above type so it is measurable. The class of all finite disjoint union of measurable rectangles is an algebra which generates the product sigma algebra; also the class of all sets $E$ in the product sigma algebra such that $P(x,E_x)$ is measurable is a monotone class. Use Monotone Class Theorem to complete the proof.