Show $X=Y$ if $E(X\mid \mathcal G) =Y$ and $E(X^2\mid\mathcal G) =Y^2$

Question

$\mathcal{G}$ is a sub-$\sigma$-algebra of $\mathcal{F}$ and $Y$ and $X$ are random variables with the property that $E(X|\mathcal{G})=Y$ and $E(X^2|\mathcal{G})=Y^2$

Show that $Y=X$ a.s.

Can you kindly tell me how to prove the equivalence?

Answer 1

$\newcommand{\E} {\mathbb E} $Hint. Consider $$\E[(X-Y) ^2] =\E[\E[(X-Y) ^2\mid\mathcal G]] $$ And also what does $\E[X\mid \mathcal G] =Y$ tell you about the measurability of $Y$?