I read this property in "Probability with Martingales" from Williams(page 91-92):
Suppose that $X_1 , ...,X_r $ are independent, each $X_k$ with law $\lambda_k$. If $h$ is bounded and $\mathcal{B}^r $-measurable ; and we define for $x_1 \in \mathbb{R} $ : $$ \gamma^h(x_1):=\mathbb{E}(h(x_1,X_2,X_3,...,X_r)) $$ then $\gamma^h(X_1) $ is a version of the conditional expectation $$\mathbb{E}(h(X_1,X_2,X_3,...,X_r)|X_1) $$
I need to figure out if the property holds when I have infinite R.V.'s $(X_k)_{k \in \mathbb{Z}} \ $ instead of $\ r $ R.V.'s ; and I condition on, say, $(X_N)_{N\in \mathbb{N}}$.
The proof for the property in the finite case uses Fubbini's Theorem; but I understand that it holds only for finite product measures.