So I know the following holds true (I can show it by using an alternative approach, and much more complex) but I would like to keep things simpler and try to find a way to prove it directly.
An old result by Cameron & Martin states that a square integrable "Brownian functional" $X\in L^2(\Omega,\mathcal F,P)$ (where $\mathcal F$ is generated by a Brownian motion) can be decomposed as a series using "generalized Hermite polynomials" (sometimes known as Wick polynomials)
$$X(\cdot)=\sum_{\alpha\in \mathcal J} x_{\alpha}\mathcal H_{\alpha}(\cdot)$$
where $\mathcal J$ is the set of multi-indices that have finitely many non-null elements and $\mathcal H_{\alpha}$ is the Wick polynomial associated with the multi-index $\alpha$. And the convergence of the series is in the $L^2$ sense.(Notice that this is just the Wiener chaos expansion in terms of Wick polynomials)
I would like to show that
$$\mathbb E [X| \mathcal G]=\sum_{\alpha\in \mathcal J} x_{\alpha}\mathbb E [\mathcal H_{\alpha}| \mathcal G]$$
where $\mathcal G$ is some sub-sigma algebra of $\mathcal F$.
So what I want to do is to interchange an integral (wrt the conditional distribution) with an $L^2$ convergent series. I wanted to apply Fubini lemma but for that I would need (this is sufficient but may be not necessary) the series to be absolutely convergent.
Is there some simple way to show that indeed one could put the conditional expectation inside the summation ? (without recurring to second quantization operators and things like that)
If $Y_n \to Y$ in $L^{2}$ then $E (Y_n|\mathcal G) \to E (Y|\mathcal G)$ in $L^{2}$ for any suib-sigma filed $\mathcal G$. This follows from conditional version of Jensen's inequality: $(E (Y_n|\mathcal G) - E (Y|\mathcal G))^{2} \leq E((Y_n-Y)^{2}|\mathcal G)$. Now take expectation on both sides to finish.