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.

**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.