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# Disintegrations are measurable functions - when are they continuous?

This is a sequel to another question I have asked.

The notion of disintegration is a refinement of conditional probability to spaces which have more structure than abstract probability spaces; sometimes this is called regular conditional probability. Let $Y$ and $X$ be two nice metric spaces, let $\mathbb P$ be a probability measure on $Y$, and let $\pi : Y \to X$ be a measurable function. Let $\mathbb P_X(B) = \mathbb P(\pi^{-1} B)$ denote the push-forward measure of $\mathbb P$ on $X$. The disintegration theorem says that for $\mathbb P_X$-almost every $x \in X$, there exists a nice measure $\mathbb P^x$ on $Y$ such that $\mathbb P$ "disintegrates":$$\int_Y f(y) ~d\mathbb P(y) = \int_X \int_{\pi^{-1}(x)} f(y) ~d\mathbb P^x(y) d\mathbb P_X(x)$$ for every measurable $f$ on $Y$.

This is a beautiful theorem, but it's not strong enough for my needs. Fix a Borel set $B \subseteq X$, and let $p(x) = \mathbb P^x(B)$. Part of the theorem is that $p$ is a measurable function of $x$. Suppose that the map $\pi : Y \to X$ is continuous instead of simply measurable. My question: What is a general sufficient condition for $p(x)$ to be continuous?

To me, this is an obvious question to ask, since if $x$ and $x'$ are two close realizations of a random $x \in X$, then the measures $\mathbb P^x$ and $\mathbb P^{x'}$ should be close too, at least in many natural situations. However, in my combing through the literature, I haven't been able to find an answer to this question. My guess is that most people are content to integrate over $x$ when they use the theorem. For my purposes, I need some estimates which I get by continuity.

At this point, I've managed to prove and write down a pretty good sufficient condition for the case I care about (Banach spaces), using an abstract Wiener space-type construction. However, I am hoping that an expert can point me toward a good reference that does this in wider generality.