 Proof of Fatou lemma

ID: proof-of-fatou-lemma

Codex  0 Created 2026-09-24 Updated 2026-09-29

g_{n} = in f_{k \geq n} f_{k} .

Then

0 \leq g_{n} ↑ lim inf_{k} f_{k}

g_{n} \leq f_{k}

for every

k \geq n

give

\int lim inf_{k} f_{k} d μ = lim_{n} \int g_{n} d μ \leq lim_{n} in f_{k \geq n} \int f_{k} d μ = lim inf_{k} \int f_{k} d μ .

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