Let $(\Omega, F, P)$ be probability space with probability measure $P$.
Theorem
Let $X\in L^1(P)$, let $F_k$ be an increasing family of sigma algebras, $F_k \subset F$ and $F=\cup_{k=1}^{\infty} \sigma(F_k)$. Then, $$E[X|F_k] \to E[X|F] \mbox{ as $k \to \infty$},$$ a.e. $P$ and in $L^1(P)$.
I want to use this theorem for $X\in L^2(P).$ First Since $X\in L^2(P)$, $X\in L^1(P)$.
So, $$E[X|F_k] \to E[X|F] \mbox{ as $k \to \infty$},$$ a.e. $P$ and in $L^1(P)$.
But, I wanna show that $E[X|F_k] \to E[X|F]$ also in $L^2(P)$.
Could you help me?
By Jensen's inequality
$$(E[X|F_n])^2 \leq E[X^2|F_n]$$
Taking Expectation
$$E(E[X|F_n])^2 \leq E[X^2]$$
Then $\sup _{n\geq 0}E(E[X|F_n])^2 < \infty$
By, Martingale $L^p$ convergence Theorem,
$E[X|F_n] \to E[X|F]$ almost surely and in $L^2$.