Let $(\Omega, \mathcal{F})$ be a measurable space, and let $\mathcal{P}$ be a collection of probability measures on this space. A sub-$\sigma$-algebra $\mathcal{G} \subset \mathcal{F}$ is said to be sufficient for $\mathcal{P}$ if for all $A \in \mathcal{F}$, there exists a $\mathcal{G}$-measurable function $\varphi_A: \Omega \to \mathbb{R}$ such that for all $P \in \mathcal{P}$,
$$
\mathbb{E}_P(1_A \mid \mathcal{G}) = \varphi_A \quad P\text{-a.s.}
$$
It is not necessary that for each $\omega \in \Omega$, the mapping $\mathcal{F} \ni A \mapsto \varphi_A(\omega)$ forms a probability measure. However, if it did, we could interpret this as describing the general distribution of $\mathcal{P}$ given $\mathcal{G}$.
Now, it is well-known that if $(\Omega, \mathcal{F})$ is a Polish space, for a fixed $P \in \mathcal{P}$, the conditional distribution $P(\cdot \mid \mathcal{G})$ can be identified with a Markov kernel. My question is whether this can be extended uniformly across all $P \in \mathcal{P}$. Specifically:
If $\mathcal{G}$ is sufficient for $\mathcal{P}$, does there exist a single Markov kernel $K: (\Omega, \mathcal{F}) \to [0, 1]$ such that for all $P \in \mathcal{P}$ and $A \in \mathcal{F}$,
$$
\mathbb{E}_P(1_A \mid \mathcal{G}) = K(A \mid \cdot) \quad P\text{-a.s.}?
$$
Bahadur (Theorem 5.1, 1954) shows this result holds when $\Omega = \mathbb{R}$ and $\mathcal{F}$ is the Borel $\sigma$-algebra. However, his proof relies heavily on properties specific to $\mathbb{R}$, such as ordering.
I suspect this result generalizes to broader contexts, such as when $(\Omega, \mathcal{F})$ is a Polish space, but I am uncertain how to adapt or generalize the proof.
Does anyone have an idea or proof strategy for this generalization? Alternatively, is there a reference where this has been worked out in detail?
Thanks in advance!
