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](https://math.stackexchange.com/questions/40436/a-question-on-measurability-in-product-spaces)  with reference to Cinlar's Probability and Stochastics, and [this paper](http://www.ams.org/journals/proc/1972-031-02/S0002-9939-1972-0291403-X/).


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



Measurability of a parametrized conditional expectation
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. 


