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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.

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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 $$\lVert E[X|\mathcal{G}]\rVert_{L^1}= E\left|E[X|\mathcal{G}]\right|\leq E\left[E[|X||\mathcal{G}]\right]=E|X|=\lVert X\rVert_{L^1}.$$

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