Consider two random variables $\tilde x$ and $\tilde y$ taking values in Polish spaces $X$ and $Y$ respectively. Let the (prior) distribution of $\tilde x$ be $\nu$ and the distribution of $\tilde y$ conditional on $\tilde x=x$ is $P_x$. So $\tilde y$ is characterized by the collection $\{({Y},{\cal B}(Y),P_x)\}_{x\in X}$.
Given $y$, then one can obtain $\mu_y$, the distribution of $\tilde x$ conditional on $y$. Then we can have another random element $\mu_{\tilde y}$ taking values in the space of distributions on $X$.
I am interested to know if $\sigma(\mu_{\tilde y}) \subset \sigma (\tilde y)$? Intuitively, it seems to be true as the information generated, or every event determined, by $\mu_{\tilde y}$ is an event that's determined by $\tilde y$. While it seems to be true with $X$ and $Y$ being finite, for general Polish spaces, I cannot approve or disapprove it. Any hint of the proof or counterexample (if it is not true in general?) Thank you!