Measurability of a parametrized conditional expectation

Question

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $\mathcal{G}\subset\mathcal{F}$ a Sub-$\sigma$-Algebra. Moreover, let $X:\Omega\rightarrow\mathbb{R}$ be a random variable and $F:\mathbb{R}^{2}\rightarrow\mathbb{R}$ a measurable function with

$$\mathbb{E}\big[\;F(X,a)\;\big] \;<\; \infty$$

for all $a\in\mathbb{R}$.

I am wondering, if we can assume the conditional expectation of $F(X,\;\cdot\;)$, if seen as a function

$$\mathbb{E}\big[\;F(X,\;\cdot\;)\;|\;\mathcal{G}\;\big] \;:\; \Omega \times \mathbb{R} \rightarrow \mathbb{R} \quad,$$

to be (jointly) $(\mathcal{G}\otimes \mathcal{B}(\mathbb{R}), \mathcal{B}(\mathbb{R}))$-measurable.

From what I understand so far, for a fixed $a\in\mathbb{R}$, the conditional expectation $\mathbb{E}\big[\;F(X,a)\;|\;\mathcal{G}\;\big]$ exists and is a $(\mathcal{G},\mathcal{B}(\mathbb{R}))$-measurable function.

Your ideas, hints and references are highly appreciated! Thank you.

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UPDATE

In case anybody else stumbles upon this question: there are readily-available results under stronger assumptions on the conditional expectation $\mathbb{E}\big[F(X,a)|\mathcal{G}\big]$, namely right-continuity in $a$.

C.f.: this question with reference to Cinlar's Probability and Stochastics, and this paper.

In any case, fedja has given the correct answer for the conditions of the original question.

Answer 1

First of all, you should be careful with how you phrase your question: the conditional expectation is defined up to a set of probability $0$, so you can (almost) always destroy the joint measurability by choosing bad slices. The best we can hope for is that we can make it measurable and that is, indeed, the case. Start with the functions of the kind $\chi_{E\times F}$ where $E$ and $F$ are measurable sets in $\Omega$ and $\mathbb R$ respectively. Then extend the result to the corresponding ring (finite unions of pairwise disjoint product sets). Then show that the sets $S\subset \Omega\times \mathbb R$ such that we can choose a jointly measurable conditional expectation for $\chi_S$ form a monotone class. This will give you the whole product $\sigma$-algebra by the monotone class lemma. After that just represent each non-negative function as an increasing limit of simple functions and any function as the difference of two non-negative functions.