Conditional dominated convergence theorem ($\mathbb{E}(X_n\mid \mathcal{G})\to \mathbb{E}(X\mid \mathcal{G})$)

If $X_n\to X$ almost surely and $|X_n|\le H$ 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 $\mathbb{E}|\mathbb{E}(X_n-X\mid \mathcal{G})|\le \mathbb{E}|X_n-X|$ 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.