# Sufficient conditions on target space for the existence of regular conditional probability

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 $X$.

The two random variables give you a probability measure $\mu$ on $\mathcal{B}(S)\otimes\mathcal{B}(T)$. It is enough to get a kernel $\kappa:S\times\mathcal{B}(T)\to[0,1]$ that reproduces $\mu$ when applied to the marginal of $\mu$ on $\mathcal{B}(S)$. Such a kernel is known as a product regular conditional probability.
A sufficient, and potentially necessary, condition is that $\mathcal{B}(S)$ is countably generated and the marginal on that space perfect in the sense of Gnedenko and Kolmogorov, or, equivalently (for e.g. $\sigma$-algebras), compact in the sense of Marczewski. This is shown in The Existence of Regular Conditional Probabilities: Necessary and Sufficient Conditions by Arnold Faden (1979).
A condition that is sufficient and necessary for countably generated probability spaces to admit only perfect probability measures is that $\mathcal{B}(S)$ is universally measurable. A topological condition that guarantees that is being a Hausdorff space that is the image of Baire space $\mathbb{N}^\mathbb{N}$ under a continuous function. A more restrictive condition, but probably the most popular, is that $S$ is Polish, that is separable and completely metrizable.
There is a weaker notion of conditional probability in which the kernel has only to be measurable with respect to the completion of the marginal on $\mathcal{B}(S)$. In this case, one can be much more general and work with spaces that are not countably generated. The seminal paper for this is Existence of Conditional Probabilities by Hoffman-Jorgensen (1971).