$X$ is a symmetric r.v, $h$ a measurable function, let $U = Xh(|X|)$, what's special about $U$?

Question

in a problem I found in some lecture notes, they ask you to use $U$ to deduce that $\mathbb{E}[X\mid |X|] = \mathbb{E}[X]$

any hints ?

Answer 1

$U$ is also a symmetric random variable, so $EU=0$ (assuming that this expectation exists). In particular $EXI_A(|X|)=0$ for any set $A$. From definition of conditional expectation this gives $E(X |\, |X|)=0=EX$.