Why is $E(E(X^2|\mathcal G))-E^2(E(X|\mathcal G))=EX^2-(EX)^2$ if $X\in \mathcal F\supset \mathcal G$
I need to prove some identity but the last step is missing, I mean $X$ is not necessarily $\mathcal G$ measurable otherwise it would be obvious, can you help
The proof rests on the fact that $\mathbb E\left[\mathbb E\left[Y\mid\mathcal A\right]\right]= \mathbb E\left[Y\right]$ for each integrable random variable $Y$ and each sub-$\sigma$-algebra $\mathcal A$.