How to prove the equality of $E[E(Y|F)^2]$ and $Y^2$? Alternatively, how to prove that conditional expectation is an orthogonal projection?

Question

I stumbled on this very interesting question that wants me to prove

$$ E[Y^2] = E[E(Y|F)^2] + E[(Y-E(Y|F))^2].$$

Which means that the conditional expectation is an orthogonal projection and it satisfies the Pythagorean identity. By expansion and simplification, I found that I just need to show that $E[E(Y|F)^2] = YE[E(Y|F)]$ or, alternatively, that $E[E(Y|F)^2] = Y^2 $. Can I simply use the law of total expectation? Thanks a lot for the help! I am really struggling with this question.

Answer 1

You don't need $E[E(Y|F)^2] = YE[E(Y|F)]$ (which is not true). What you need is the fact that both sides of this have the same expectation: $E[E[E(Y|F)^2]] = E[YE[E(Y|F)]]$. To prove this start with RHS and use the tower property. When you condition on $F$ you get $E[YE[E(Y|F)]]=E[ E[YE[E(Y|F)]]|F]$. Now pull out the the inside $E(Y|F)$ : It is already measuarble w.r.t. $F$). Now you get LHS easily.