A question about conditional expectations: $E[G(\xi)|F(\xi, \eta)]=E[G(\eta)|F(\xi, \eta)]$

Question

$\xi$, $\eta$ are random variables, and they are independent and identically distributed. $F(x, y)$ is a Borel function such that $F(x, y) = F(y,x)$. $G$ is a Borel function such that $G(\xi)$ is integrable. Prove that $$E[G(\xi)|F(\xi, \eta)]=E[G(\eta)|F(\xi, \eta)].$$ Can someone give me some hints? Thanks.

Answer 1

You have to show that $EG(\xi )I_{F(\xi, \eta )^{-1} (B)} =EG(\eta )I_{F(\xi, \eta )^{-1} (B)}$ for every Borel set $B$ in $\mathbb R$. Simply write both sides as integrals w.r.t. the joint distribution of $\xi$ and $\eta$. By the hypothesis $(\xi, \eta )$ has the same joint distribution as $( \eta , \xi)$. If you just switch the variables in the integral on the left side you will get the right side.