Let $X$ be a square-integrable random variable on $(\Omega, \mathcal{A}, P)$. Let $\mathcal{F}$ be a sub-sigma-Algebra and let $Z$ denote $Z:=E[X \mid \mathcal{F}]$.
We know, since the conditional expectation is an orthogonal projection, that $X-Z$ is orthogonal to $Z$.
Moreover we know that $X-Z$ is orthogonal to 1, since they have the same expectation.
Now we have written:
$$E[(X-E[X])^2]=E[((X-Z) + (Z-E[X]))^2] = E[(X-Z)^2] + E[(Z-E[X])^2]$$
by Pythagoras' Theorem. Can you state Pythagoras' Theorem for random variables (I could not find it) and explain why/how it applies here? And which assumptions are needed for that.
Thanks a lot!