Expected value of square of conditional expectation $E[(E[X|\mathcal{F}])^2]$

Question

Say I have a random variable $X$ with expected value zero $E[X]=0$, I now wish to prove that \begin{equation} E[(E[X|\mathcal{F}])^2] = E[X^2] \end{equation} where $\mathcal{F}$ is some subalgebra. Intuitively this makes sense since the expected variance of an estimate should equal the variance of what is being estimated, but I'm not sure how to prove this. Any suggestions?

Answer 1

This is not true! In fact, the expectation operator is an $L^2$ contraction i.e. one does have that $E[E(X |\mathcal F)^2] \leq E[X^2]$ (via the conditional Jensen inequality) but equality will not hold.

For example, any random variable $Z$ with non-zero variance gives non-equality with $\mathcal F = \{\emptyset,\Omega\}$ the trivial sigma - algebra, because then $E[Z|\mathcal F] = E[Z]$ so that $E[E[Z | \mathcal F]^2] = E[Z]^2$ which will be strictly smaller than $E[Z^2]$.

Answer 2

If $X$ is independent of $\mathcal F$ the LHS is $0$ so the equation is not correct. LHS $\leq $ RHS is always true.