Conditional Expectation of $XY$ given $X^2+Y^2$ Suppose $X$ and $Y$ are two iid random variables in $L^2$ with symmetric distributions (like Gaussian mean $0$). Prove that $E[XY \mid X^2 + Y^2] = 0$ a.s.
This makes sense intuitively, and I was able to verify it computationally, sampling $X$ and $Y$ from $N(0,1)$. But how do I prove this from the definition of conditional expectation? In other words, is it true that for all $A \in \sigma(X^2+Y^2)$, $\int_A XY \, dP = 0$?

We can write $A$ as $(X^2+Y^2)^{-1}(B)$ for some Borel set $B$. Now $\int_A XY\,dP=\int_C xy \,dF(x)\,dF(y)$ wher $C=\{(x,y): x^2+y^2 \in B\}$ and $F$ is the common  distribution.  Make the change of variable $(x,y) \to (-x,y)$ to see that this integral is $0$.
