Monotone convergence theorem in the proof of the pythagorean theorem in conditional expectation

Question

Assume $X$ is a $L^2$ random variable on $(\Omega, \mathscr{F}, P)$, $\mathscr{G}$ is a sub-sigma-algebra, then consider the conditional expectation $E[X|\mathscr{G}]$. My professor suggested that $E[X|\mathscr{G}]$ is the best approximation of $X$ in the sense that it is the $\mathscr{G}$-measurable random variable $Y$ which minimizes $E[(X-Y)^2]$. Suppose $Y$ is a $\mathscr{G}$-measurable random variable. In his proof, he wrote: \begin{align*} E[(X-Y)^2]&=E[(X-E[X|\mathscr{G}]+E[X|\mathscr{G}]-Y)^2]\\ &=E[(X-E[X|\mathscr{G}])^2] + E[(Y-E[X|\mathscr{G}])^2]+2E[(X-E[X|\mathscr{G}])(E[X|\mathscr{G}]-Y)]\\ &=E[(X-E[X|\mathscr{G}])^2] + E[(Y-E[X|\mathscr{G}])^2] \end{align*} He mentioned that actually for any $\mathscr{G}$-measurable random variable $Z$, we always have $E[(X-E[X|\mathscr{G}])Z] = 0$, suggesting that first consider $E[(X-E[X|\mathscr{G}])1_A] = 0$ where $A\in \mathscr{G}$, then use the monotone convergence theorem. I know that it derives $E[(X-E[X|\mathscr{G}])f(\omega)] = 0$ for any $\mathscr{G}$-measurable simple function, then use it to approximate a general function. But I cannot see how to use the monotone convergence theorem here. $X-E[X|\mathscr{G}]$ is not always positive here, then how to use the theorem? I have difficulty writing down the detail of the proof. Thank you!

Answer 1

I would write a comment, but I cannot. If I understand your question correctly, you have \begin{align} \mathbb{E}[(X-\mathbb{E}[X|\mathcal{G}])Z_s]=0 \end{align} for any simple $Z_s \in L^1(\Omega,\mathcal{G},\mathbb{P})$ and want to show \begin{align} \mathbb{E}[(X-\mathbb{E}[X|\mathcal{G}])Z]=0 \end{align} for any nonnegative $\mathcal{G}$ measurable random variable $Z$ in $L^1$? As you seem to know, you can find a sequence $(Z_n)_{n\in \mathbb{N}}$ such that $Z_n$ converges pointwise from below to $Z$. Then write \begin{align} \mathbb{E}[(X-\mathbb{E}[X|\mathcal{G}])Z_n] &=\mathbb{E}[((X-\mathbb{E}[X|\mathcal{G}])^+ -(X-\mathbb{E}[X|\mathcal{G}])^-) Z_n] \\ &=\mathbb{E}[\underbrace{(X-\mathbb{E}[X|\mathcal{G}])^+ Z_n}_{\geq 0}]-\mathbb{E}[\underbrace{(X-\mathbb{E}[X|\mathcal{G}])^-Z_n}_{\geq 0} ], \end{align} where I used the notation $a^+=\max\{a,0\}, a^-=\max\{-a,0\}$. You can then apply the monotone convergence theorem to each single term and conclude as usual.