I asked this question in mathstackexchange a couple of days ago. Almost right after posing it a partial (affirmative) answer came to my mind in the following form

**Proposition:** *Assume that $(X,\mathcal{X},\mu)$ and $(Y,\mathcal{Y},\nu)$ are probability spaces and that $f\in L^{1}_{\mu\times\nu}$. Then for every countably generated $\sigma-$field $\mathcal{X}_{0}\subset \mathcal{X}$ and any version $E[f|\mathcal{X}_{0}\times\mathcal{Y}]$ of the conditional expectation of $f$ with respect to $\mathcal{X_{0}}\times\mathcal{Y}$, there exists $Y'\subset Y$ with $\nu(Y')=1$ such that for all $y\in Y'$ $E[f|\mathcal{X}_{0}\times\mathcal{Y}](\cdot,y)$ is a version of $E[f(\cdot,y)|\mathcal{X}_{0}]$.*

While I thought that this solved my particular problem (which has to do with the proof of a quenched functional limit theorem for Fourier Transforms in which I am working), I realized later that this proposition is not enough for my needs. Rather I need to know whether the following -the second version of the question in stackexchange- is true:

**Question:** Assume that $E[\,\cdot\,|\mathcal{X}_{0}]$ admits a *regular* version. This is: there exists a family of measures $\{\mu_{x}\}_{x\in X}$ such that for every $g\in L^{1}_{\mu}$
$$x\mapsto \int_{X}g(z)d\mu_{x}(z)$$
defines a version of $E[g|\mathcal{X}_{0}]$. Is the function
$$(x,y)\mapsto \int_{X}f(z,y)d\mu_{x}(z)\,\,\,\,\,\,\mbox{(1)}$$
(which is $\mu-$a.e well defined for $\nu-$a.e $y$ by the integrability of $x\mapsto f(x,y)$ for $\nu$-almost every $y$) extensible to a $\mathcal{X}\times \mathcal{Y}$ measurable map, or even to a $\mathcal{X}_{0}\times \mathcal{Y}$ measurable map? (in other words for this last case, can we think of (1) as a version of $E[f|\mathcal{X}_{0}\times\mathcal{Y}]$?).

Two comments:

**1.** I am posing the question here because it did not go very far in Stackexchange. Indeed it was voted down a couple of times. If you think that the question does not deserve an answer I would appreciate any reference or comment explaining why (is it trivial?, please say why in such case. Is the answer well known? please give a reference then...).

**2.** (For your curiosity) I need a (positive) answer to this question in order to ''integrate'' certain quenched results to obtain a quite general (expected) limit theorem for the Fourier transforms

$$S_{n}(\theta,\omega)=\sum_{k=0}^{n-1}X_{k}(\omega)e^{ik\theta}$$ of an ergodic process $(X_{k})_{k}$ in $L^{2}$, but I'm stuck at a certain step related to this. Before giving up (or looking for an alternative statement to my ''theorem'') I'd like to se if anyone can help.

Thanks for your attention!