Is $E(XY) = E(XE(Y|Z))$ true?

Question

Is $E(XY) = E(XE(Y|Z))$ always true if I don't know the relationship between these 3 random variables.

I think when $\sigma(X) \subset \sigma(Z)$, the equation is true, but what will happen otherwise?

Answer 1

Suppose $X$ is $+1$ or $-1$ with equal probability, $Y=X$, and $Z$ is anything independent of $X$ and $Y$.

Then $E(XY) = E(X^2) = 1$. On the other hand, $E(Y|Z) = E(Y) = 0$, so $E(XE(Y|Z)) = 0$ as well.