= Conditional Fatou lemma
{c}
For nonnegative <random variables> $X_n$ on a <probability space> and a sub-<sigma-algebra> $\mathcal G$,
$$
\mathbb E[\liminf_nX_n\mid\mathcal G]\leq\liminf_n\mathbb E[X_n\mid\mathcal G]
$$
almost surely, with extended nonnegative conditional expectations allowed. Use one common full-probability set for the countably many conditional inequalities.
Put $Y_n=\inf_{k\geq n}X_k$. Then $0\leq Y_n\uparrow\liminf_kX_k$, and <monotonicity> of <conditional expectation> gives
$$
\mathbb E[Y_n\mid\mathcal G]\leq\inf_{k\geq n}\mathbb E[X_k\mid\mathcal G].
$$
Conditional <monotone convergence> gives the limit of the left side, proving the claim. That conditional convergence statement follows by integrating the increasing conditional expectations on every event in $\mathcal G$ and applying ordinary <monotone convergence>.
Testing the ordinary <Fatou lemma> on events alone is not the whole argument: it bounds an integral by a lower limit of integrals, which is not automatically the integral of the conditional lower limit. The increasing tail infima supply the needed order comparison.
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