Prove $\mathbb E(X∣Y)=0$ for $2$ variables

Question

I was looking for an example of two dependent random variables in which
$\mathbb E(X|Y)=\mathbb E(X)$

I found this example:
$X∼U[−1,1]$ and $Y=X^2$

---

How can I prove that $\mathbb E(X∣Y)=0$?

thanks!

Answer 1

$E(XI_{{X^{2}} \leq x})=0$ for each $x$ because $X$ has same distribution as $-X$. This implies that $E(XI_A)=0$ for any $A \in \sigma (Y)$ and hence $E(X|Y)=0$.

Answer 2

Hint: $E(X|X^2) = E(-X|X^2) = -E(X|X^2)$