# Convergence of conditional expectations given convergence of random variables

Say $$X_n$$ is an $$\mathcal{F}_n$$-adapted sequence of random variables converging to $$X_{\infty}$$ in $$L^1$$. Clearly, fixed $$n\leq m$$ one has $$E[X_m\mid \mathcal{F}_n] \to E[X_{\infty}\mid \mathcal{F}_n]$$ in $$L^1$$. What's the easiest way to prove this? I know that $$E[X_{\infty}\mid \mathcal{F}_n]\in L^1$$ but I can't use the dominated convergence because I don't have P-a.s. convergence of the random variables, nor monotone convergence because I don't have non-negativity.

This amounts to saying that the mapping $$X \mapsto E[X|\mathcal{G}]$$ is continuous for the $$L^1$$ norm (here $$\mathcal{G}$$ is any sub-$$\sigma$$-field.)
The mapping is clearly linear and satisfies $$\lVert E[X|\mathcal{G}]\rVert_{L^1}= E\left|E[X|\mathcal{G}]\right|\leq E\left[E[|X||\mathcal{G}]\right]=E|X|=\lVert X\rVert_{L^1}.$$