1. ## conditional expectation

If $E(X^2)< \infty$, then $E((X-E(X|G))^2) \leq E((X-E(X))^2)$

2. Start with $E((X-E(X))^2)$ and add and subtract $\mu_g=E(X|G)$

$E((X-E(X))^2)=E((X-\mu_g+\mu_g-E(X)^2)$

$=E((X-\mu_g)^2)+2E((X-\mu_g)(\mu_g-\mu))+E(\mu_g-\mu)^2$ where $\mu=EX$

$=E((X-\mu_g)^2)+E(\mu_g-\mu)^2\ge E((X-\mu_g)^2)$

Since

$E((X-\mu_g)(\mu_g-\mu))=E(E((X-\mu_g)(\mu_g-\mu)|G))$

$=E ((\mu_g-\mu)(E(X-\mu_g) |G))=E((\mu_g-\mu)(\mu_g-\mu_g))=0$