Suppose $H:\Omega\times X\mapsto Y$ for some borel subset $X\subset \mathbf{R}$, Euclidean space $Y$, and probability space $(\Omega, \mathcal{F},P)$. Further suppose that $H$ is $C^1$ for each fixed $\omega\in\Omega$, and measurable for each $x\in X$.
Let $\mathcal G\subset \mathcal{F}$ be a sub-$\sigma$-algbebra, and let $\hat E$ denote the regular conditional expectation in the sense of Dynkin and Evstigneev or the conditional expectation of random integrands in the sense of Castaing and Ezzaki.
How do we know
1) If $\frac{\partial H}{\partial x}\hat E[H(\cdot, x)|\mathcal G]$ exists for each $x$ a.s.
2)When is $$\frac{\partial H}{\partial x}\hat E[H(\cdot, x)|\mathcal G]=\hat E[\frac{\partial H}{\partial x}H(\cdot, x)|\mathcal G]$$ a.s.
3) When is $$\frac{\partial H}{\partial x}\hat E[H(\cdot, x)|\mathcal G]_{x=u(\cdot)}=\hat E[\frac{\partial H}{\partial x}H(\cdot, u(\cdot))|\mathcal G]$$ a.s whenever $u:\Omega\rightarrow X$ is $\mathcal G$-measurable?
$\{|\partial H/ \partial x(\cdot,x)| : x \in X\}$are uniformly integrable, then this follows quickly from the dominated convergence theorem for conditional expectations. The details are essentially the same as here: math.stackexchange.com/questions/94628/… $\endgroup$