I have two questions for my assignment that I don't know how to prove.
Both are based on the condition (omega, F, P) is a probability space, and G is a sub sigma-field of F.
The first question is show that E(XE[X|G]) = E[E[X|G)^2].
The second question is Let X be a F-measuarable random variable such that E(X^2) is finite. Prove that E(E[X|G] - X)^2 = inf {E(Y-X)^2, E(Y^2)
They are about conditional expectation of random variables. Can anyone please help？Thanks a lot!
Hint: (1) $E[X\mid \mathcal G]$ is $\mathcal G$-measurable, hence $$ \def\G{\mathcal G}E\bigl[XE[X\mid \G] \mid \G\bigr] = E[X \mid \G]E[X\mid G] $$ now take expected values and use that $E[E[Y\mid G]] = E[Y]$ for any $Y$.
(2) We havefor $Y \in L^2(\mathcal G)$, $\def\abs#1{\left|#1\right|}$ \begin{align*} E\bigl[Y(X-E[X\mid G])\bigr] &= E\bigl[XY - E[XY\mid G]\bigr]\\ &= E[XY]- E[XY]\\ &= 0, \end{align*} Hence, for any $Y \in L^2(\mathcal G)$: \begin{align*} E\bigl[\abs{X-Y}^2\bigr] &= E\bigl[\abs{X-E[X\mid G] - (Y - E[X\mid G])}^2\bigr]\\ &= E[\abs{X-E[X\mid G]}^2] - 2E[(X-E[X\mid G])(Y-E[X\mid G])] + E[\abs{Y-E[X\mid G]}^2]\\ &= E[\abs{X-E[X\mid G]}^2] + E[\abs{Y-E[X\mid G]}^2] \end{align*}