= Conditional dominated convergence theorem
{title2=$\mathbb E(X_n\mid\mathcal G)\to\mathbb E(X\mid\mathcal G)$}
If $X_n\to X$ almost surely and $|X_n|\leq H$ for integrable $H$, <conditional expectations> converge almost surely and in $L^1$. <Conditional Fatou lemma> applied to $H+X_n$ and $H-X_n$ bounds the conditional liminf and limsup by the same limit. For $L^1$ convergence, the contraction estimate $\mathbb E|\mathbb E(X_n-X\mid\mathcal G)|\leq\mathbb E|X_n-X|$ and <dominated convergence theorem> finish the proof. This also permits taking an unbounded measurable factor out of a <conditional expectation> by clipping it, provided the original product is integrable.
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