Let $\mu:\mathbb{R}\times \mathscr{B}_{\mathbb{R}}\rightarrow [0,1]$ be a Markov kernel. (That is $\mu_x$'s are probability measures and $\mu^A$'s are Borel measurable)
Then, does there exist random variables $X,Y$ such that $\mu^A\circ Y$ is a version of $E(\mathbb{1}_{X\in A} | Y)$ for all $A$?
(One can prove the converse. If random variables $X,Y$ are given (taking values in Polish space), then there exists a Markov kernel $\mu$ such that $\mu^A\circ Y$ is a version of $E(\mathbb{1}_{X\in A} | Y)$ for all $A$. I am curious whether all Markov kernels can be induced in this way)