Product measurability of the Radon-Nikodym derivative of two Marcov kernels

Question

Let $(\mathcal{X},\mathcal{A})$ and $(\mathcal{Y},\mathcal{B})$ are Borel spaces.

Let $\kappa_i: \mathcal{B} \times \mathcal{X} \to [0,1], ~i=1,2$ be Marcov kernels from $(\mathcal{X},\mathcal{A})$ to $(\mathcal{Y},\mathcal{B})$, i.e., for each $B \in \mathcal{B}$, $x \mapsto \kappa_i(B, x)$ is a $\mathcal{A}$-measurable function, and for each $x \in \mathcal{X}$, $B \mapsto \kappa_i(B, x)$ is a probability measure on $(\mathcal{Y}, \mathcal{B})$.

Suppose that for any $x \in \mathcal{X}$, $\kappa_1(\cdot, x) \ll \kappa_2(\cdot, x)$ and let $f(x,y)$ denote the Radon-Nikodym derivative $\frac{d\kappa_1(\cdot, x)}{d\kappa_2(\cdot, x)}(y)$.

My question: is it true that $(x,y) \mapsto f(x,y)$ is $\mathcal{A} \otimes \mathcal{B}$-measurable?

I am not familiar with the Marcov kernel, but this question is motivated by that I want to resolve my previous question.

Any comments are welcome. Thank you very much.

RND's are only defined almost everywhere, so you cannot expect measurability. — Kavi Rama Murthy, Dec 16 '22 at 05:01
@geetha290krm Thank you so much for your comment. What do you mean by "only defined almost everywhere"? For each $x \in \mathcal{X}$, the RND $y \mapsto f(x,y)$ is $\mathcal{B}$-measurable by definition. — Sugiyama, Dec 16 '22 at 05:40
Kallenberg (2002) Proposition 7.26 (regular densities) is the solution if the kernels are of the form $\kappa_1(\cdot, x) = \mu(\cdot \times B, x)$ and $\kappa_2(\cdot, x) = \mu(\cdot \times U, x)$ for some $B\subseteq U$ — Felix Benning, Mar 07 '25 at 14:23
The solution and proof can be found in E. Çinlar - Probability and Stochastics Chapter 5 Theorem 4.44. I've seen it called "Doob's version of the RN theorem". The statement is roughly: if $\mathcal{B}$ is separable, then there exists $f(x,y)$ jointly $\mathcal{A} \otimes \mathcal{B}$ measureable so that $f(x, \cdot) = \frac{d \kappa_1(\cdot, x)}{d \kappa_2(\cdot, x)}$ (up to $\kappa_2(\cdot, x)$-null modifications).
The condition that $\mathcal{B}$ is separable is critical. — jet457, Mar 23 '25 at 04:18

Product measurability of the Radon-Nikodym derivative of two Marcov kernels

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