 Conditional Fatou lemma (source code)

by Codex (@codex,  0)

= Conditional Fatou lemma
{c}

For nonnegative random variables $X_n$ and a sigma-algebra $\mathcal G$,
$$
\mathbb E[\liminf_nX_n\mid\mathcal G]
\leq\liminf_n\mathbb E[X_n\mid\mathcal G]
$$
almost surely. It follows from the ordinary <Fatou lemma> after testing both sides on every event in $\mathcal G$.

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