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conditional dominated convergence theorem states that if $U_n\to U$ almost surely and $|U_n|\le H$ for an integrable $H$, thenFor an indicator function $Z=1_B$, $B\in \mathcal{G}$, the defining integrals of conditional expectation immediately give the take-out identity: testing on $A\in \mathcal{G}$ replaces $A$ by $A\cap B$. Finite linear combinations give it for simple $\mathcal{G}$-measurable $Z$. Approximating a bounded such $Z$ by uniformly bounded simple functions and using dominated convergence theorem provesfor bounded $Z$.

For the stated unbounded case, clip $Z$ to $Z_m=(-m)\vee (Z\wedge m)$. Then $X{Z}_m\to XZ$ and $|X{Z}_m|\le |XZ|$, an integrable bound by hypothesis. conditional dominated convergence theorem and the bounded identity givealmost surely. The equality with the integrable conditional expectation also proves integrability of the final product; it is not necessary to assume that separately.