Let $\mathcal{F}$ and $\mathcal{G}$ be two σ-fields and suppose $\mathcal{F}\subset\mathcal{G}$. Let $X$ be a $\mathcal{F}$-measurable random variable. Show that $E(X^2)\ge E(E(X \mid \mathcal{G})^2)$.
I can work it out with real numbers but I have no idea how to prove it. Thank you for helping.
If $Y$ is $\mathcal G$-measurable then: $$\mathbb EXY=\mathbb E[\mathbb E[XY\mid\mathcal G]]=\mathbb E[\mathbb E[X\mathcal\mid G]Y]$$
Applying that for $Y=\mathbb E[X\mid\mathcal G]$ we find: $$\mathbb E[X\mathbb E[X\mid\mathcal G]]=\mathbb E[\mathbb E[X\mid\mathcal G]^2]\tag1$$
Of course: $$\mathbb E(X-\mathbb E[X\mid\mathcal G])^2\geq0$$
Working out the RHS we find with $(1)$: $$\mathbb E(X-\mathbb E[X\mid\mathcal G])^2=\mathbb EX^2-\mathbb E[\mathbb E[X\mid\mathcal G]^2]$$