Conditional expectation on product of two probability spaces Assume that we have probability spaces $\left(\Omega_i,\mathcal{A}_i,\mathcal{P}_i\right)$ for $i=1,2$. We form the probability space $\left(\Omega_1\times\Omega_2,\mathcal{A}_1\otimes\mathcal{A}_2,\mathcal{P}_1\times\mathcal{P}_2\right)$. What can we say about the conditional expectation
$$
E_{\mathcal{P}_1\times\mathcal{P}_2}[X|\mathcal{A}_1'\otimes\mathcal{A}_2']
$$
for $\mathcal{A}_i'\subset\mathcal{A}_i$ and $X:\left((\Omega_1\times\Omega_2),(\mathcal{A}_1\otimes\mathcal{A}_2))\rightarrow(\mathbb{R},\mathcal{B}(\mathbb{R}))\right)$. 
Is it possible to reduce it in some cases to the conditional expectation on the underlying spaces?
 A: I claim that 
$$
\mathbf{E}_{\mathcal{P}_1\times\mathcal{P}_2}[X\mid\mathcal{A}_1'\otimes\mathcal{A}_2'] = \mathbf{E}_{\mathcal{P}_2}[\mathbf{E}_{\mathcal{P}_1}[X\mid\mathcal{A}_1']\mid\mathcal{A}_2']=:\eta,
$$
where in the rhs the first expectation is taken w.r.t. $\omega_1$, the second, w.r.t. $\omega_2$. To this end, we need to show that the rhs is $\mathcal{A}_1'\otimes\mathcal{A}_2'$-measurable (I'll leave this for you) and that 
$$
\mathbf{E}_{\mathcal{P}_1\times\mathcal{P}_2}[X\mathbf{1}_A] = \mathbf{E}_{\mathcal{P}_1\times\mathcal{P}_2}[\eta\mathbf{1}_A]
$$
for any $A\in\mathcal{A}_1'\otimes\mathcal{A}_2'$. By a monotone class argument, it is enough to show this for $A = A_1\times A_2$, where $A_1\in \mathcal{A}_1'$, $A_2\in\mathcal{A}_2'$. Using the Fubini theorem,
$$
\mathbf{E}_{\mathcal{P}_1\times\mathcal{P}_2}[\eta\mathbf{1}_{A_1\times A_2}] =  \mathbf{E}_{\mathcal{P}_1}[\mathbf{E}_{\mathcal{P}_2}[\eta\mathbf{1}_{\omega_1\in A_1}\mathbf{1}_{\omega_2\in A_2}]]\\
= \mathbf{E}_{\mathcal{P}_1}\big[\mathbf{E}_{\mathcal{P}_2}[\mathbf{E}_{\mathcal{P}_2}\big[\mathbf{E}_{\mathcal{P}_1}[X\mid\mathcal{A}_1']\mid\mathcal{A}_2']\mathbf{1}_{\omega_1\in A_1}\mathbf{1}_{\omega_2\in A_2}\big]\big] \\
= \mathbf{E}_{\mathcal{P}_1}\big[\mathbf{E}_{\mathcal{P}_2}[\mathbf{E}_{\mathcal{P}_2}\big[\mathbf{E}_{\mathcal{P}_1}[X\mid\mathcal{A}_1']\mid\mathcal{A}_2']\mathbf{1}_{\omega_2\in A_2}\big]\mathbf{1}_{\omega_1\in A_1}\big]\\
=  \mathbf{E}_{\mathcal{P}_1}\big[\mathbf{E}_{\mathcal{P}_2}[\mathbf{E}_{\mathcal{P}_1}[X\mid\mathcal{A}_1']\mathbf{1}_{\omega_2\in A_2}\big]\mathbf{1}_{\omega_1\in A_1}\big]\\
 =  \mathbf{E}_{\mathcal{P}_2}\big[\mathbf{E}_{\mathcal{P}_1}[\mathbf{E}_{\mathcal{P}_1}[X\mid\mathcal{A}_1']\mathbf{1}_{\omega_1\in A_1}\big]\mathbf{1}_{\omega_2\in A_2}\big]\\
=  \mathbf{E}_{\mathcal{P}_2}\big[\mathbf{E}_{\mathcal{P}_1}[X\mathbf{1}_{\omega_1\in A_1}\big]\mathbf{1}_{\omega_2\in A_2}\big]
= \mathbf{E}_{\mathcal{P}_1\times \mathcal{P}_2}\big[X\mathbf{1}_{A_1\times A_2}\big],
$$
as required.
