TITLE: Convergence of conditional second moments
QUESTION [5 upvotes]: Let $(\Omega, \mathcal{A},P)$ be a probability space, and let $(\mathcal{F}_k)_{k \geq 1}$ be a filtration which converges to $\mathcal{A}$. I suppose it is true that 
$$
E \left( \big(E \left( X | \mathcal{F}_k \right) \big)^2 \right) \to E \left(X^2 \right).
$$
How to prove this? I guess one will need Jensen's inequality and a convergence theorem for submartingales, but cannot find the right reference (I am working on number theory, not probability).
(If you know an answer, please also provide a citable reference.)

REPLY [5 votes]: Let us state Corollary 2.1 of these notes.

Let $p>1$, $X\in\mathbb L^p$ and let $\left(\mathcal F_n\right)_{n\geqslant 1}$ be a filtration. Denote by $\mathcal F$ the $\sigma$-algebra generated by $\bigcup_{n\geqslant 1}\mathcal F_n$. Then the convergence 
  $$\lim_{n\to +\infty} \mathbb E\left[X\mid\mathcal F_n\right]=\mathbb E\left[X\mid\mathcal F\right],$$
  where the convergence holds almost everywhere and in $\mathbb L^p$.

The wanted result follows from this corollary and the elementary fact that if $\left\lVert Y_n-Y\right\rVert_2\to 0$ then $\left\lVert Y_n\right\rVert_2^2\to \left\lVert Y\right\rVert_2$.