I would like to ask about the measurability of the conditional expectation of $Z$ given $X=x$ calculated on the conditional probability measure $\mathbb{P}_{X,Z|Y=y}$.
Assumptions
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $(\mathcal{X}, \mathcal{A})$ and $(\mathcal{Y}, \mathcal{B})$ be measurable spaces such that the existence of a regular conditional distribution is guaranteed (e.g., Euclidean space). Let $X: \Omega \to \mathcal{X}$, $Y: \Omega \to \mathcal{Y}$ and $Z: \Omega \to \mathbb{R}$ are random quantities (i.e., measurable mappings) such that $\mathbb{E}[\lvert Z \rvert] < \infty$.
For each $y \in \mathcal{X}_2$, let $h(x,y) = \mathbb{E}_{(X,Z) \sim \mathbb{P}_{(X,Z)|Y=y}}[Z|X=x]$, which denote the conditional expectation of $Z$ given $X = x$ calculated in the probability space $(\mathcal{X} \times \mathbb{R}, \mathcal{A} \otimes \mathcal{B}(\mathbb{R}), \mathbb{P}_{X,Z|Y=y})$, where $\mathbb{P}_{X,Z|Y=y}$ is a (regular) conditional distribution of $(X,Z)$ given $Y=y$.
My question
Is the following proposition true? And if it is true, could you tell me how to show it?
$(x,y) \mapsto h(x,y)$ is a $(\mathcal{A} \otimes \mathcal{B})$-measurable function.
In Double conditional probability, this proposition is assumed, but according to the proof of Theorem B.75. (on page 633) of Theory of Statistics, by Mark J. Schervish, it is true.