 Conditional Fatou lemma

For nonnegative random variables $X_n$ on a probability space and a sub-sigma-algebra $\mathcal{G}$,almost surely, with extended nonnegative conditional expectations allowed. Use one common full-probability set for the countably many conditional inequalities.

Put $Y_n={\inf }_{k\ge n}{X}_k$. Then $0\le Y_n\uparrow {\mathrm{liminf}}_k{X}_k$, and monotonicity of conditional expectation givesConditional 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.