Tell me more ×  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