Suppose $(\Omega, \mathcal{F}, \mathbb{P})$ is a probability space, $(S, \mathcal{B}(S))$ and $(T, \mathcal{B}(T))$ are topological spaces with their Borel $\sigma$-algebras, and $X: \Omega \to S$ and $Y: \Omega \to T$ are random variables.
I know there are conditions I can put on $(\Omega, \mathcal{F}, \mathbb{P})$ to guarantee I can find regular conditional probabilities for an arbitrary random variable and measurable map. I'm wondering whether there are topological conditions I can put on $S$ and $T$ which guarantee that there exists a regular conditional probability for $Y$ given $X$.
By regular conditional probability, I mean a map $\nu: S\times \mathcal{B}(T) \to [0,1]$ such that: (1) For each $s \in S$, $\nu(s, \cdot)$ is a probability measure on $(T, \mathcal{B}(T))$, (2) For each $B\in \mathcal{B}(T)$, $\nu(\cdot, B)$ is measurable, (3) For each $A\in \mathcal{B}(S), B\in \mathcal{B}(T)$, $\mathbb{P}\{X\in A, Y\in B\} = \int_A \nu(\cdot,B)d\mathbb{P}_X$. Where $\mathbb{P}_X$ is the pushforward probability measure of $X$.