Conditional Expectation of a product of r.v.

Question

I am exploring a work in probability theory that states $$E[U1U2 | V1, V2] = E[U1| V1]E[ U2 | V2]$$ for independent pairs of random variables $(U1,V1)$ and $(U2,V2)$. I understand the intuition behind this equation, but I'm looking for a rigorous proof or a reference that provides a detailed explanation. Could anyone provide a proof (or sketch of proof) or point me towards a comprehensive resource that proves this equality, especially under the assumption of independence of these variable pairs?

Answer 1

The random variable $\mathbb E\left[U_1\mid V_1\right]\mathbb E\left[U_2\mid V_2\right]$ is $\sigma(V_1,V_2)$-measurable hence it suffices to check that for each $C\in \sigma(V_1,V_2)$, $$\tag{*} \mathbb E\left[U_1U_2\mathbf{1}_{C}\right]=\mathbb E\left[\mathbb E\left[U_1\mid V_1\right]\mathbb E\left[U_2\mid V_2\right]\mathbf{1}_{C}\right]. $$ Since $\sigma(V_1,V_2)$ is generated by the $\pi$-system of sets of the form $\{V_1\in A\}\cap \{V_2\in B\}$, where $A,B$ are Borel subsets of the real line, it suffices to prove $(*)$ when $C$ is of the form $\{V_1\in A\}\cap\{V_2\in B\}$, which can be done as follows \begin{align} \mathbb E\left[U_1U_2\mathbf{1}_{C}\right]&=\mathbb E\left[U_1U_2\mathbf{1}_{V_1\in A}\mathbf{1}_{V_2\in B}\right]&\\ &=\mathbb E\left[U_1\mathbf{1}_{V_1\in A}\right]\mathbb E\left[U_2\mathbf{1}_{V_2\in B}\right]&\mbox{ by independence between }(U_1,V_1)\mbox{ and }(U_,V_2)\\ &=\mathbb E\left[\mathbb E\left[U_1\mid V_1\right]\mathbf{1}_{V_1\in A}\right]\mathbb E\left[\mathbb E\left[U_2\mid V_2\right]\mathbf{1}_{V_2\in B}\right]&\mbox{by definition of conditional expectation}\\ &=\mathbb E\left[\mathbb E\left[U_1\mid V_1\right]\mathbf{1}_{V_1\in A} \mathbb E\left[U_2\mid V_2\right]\mathbf{1}_{V_2\in B} \right] &\mbox{ by independence between }V_1\mbox{ and }V_2\\ &=\mathbb E\left[\mathbb E\left[U_1\mid V_1\right]\mathbb E\left[U_2\mid V_2\right]\mathbf{1}_{C}\right]. \end{align}