Expectation how does $E[XY^2]=E[Y^2E[X|Y]]$?

Question

Given random variables X and Y show that $E[XY^2]=E[Y^2E[X|Y]]$

For the case that $X$ and $Y^2$ are independent I have $$E(XY^2)=E(X)(E(Y^2)= E(E(X|Y))E(Y^2)=E(E(X|Y)Y^2)$$ but I'm sure about the general case. Any help is appreciated.

Answer 1

By the tower property of conditional expectation plus measurability of the random variable $Y$ given its own value.

$$ E[XY^2] =\text{(tower)}= E[E[XY^2|Y]] = (\text{measurable}) = E[Y^2E[X|Y]] $$