LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Fatou's lemma

Statement

Let $(X, Σ, μ)$ be a measure space and let $f_{n} : X \to [0, \infty]$ be measurable. Then $\int {lim inf}_{n \to \infty} f_{n} d μ \leq {lim inf}_{n \to \infty} \int f_{n} d μ$ .

Facts & Assumptions

Given: As stated in the item.

[F1]

Measure space and integral: nonnegative integrals are monotone and the integral of a nonnegative measurable function is defined by simple minorants.

[F2]

Monotone convergence for the measure integral: integrals commute with increasing pointwise limits, including infinite values.

[A1]

The sequence $f_{n}$ is nonnegative and measurable on the stated measure space.

Proof

Given: A measure space and measurable $f_{n} \geq 0$ .

1.1F1A1

Set $h_{k} : = \inf_{n \geq k} f_{n}$ . For each real $a$ , ${h_{k} < a} = ⋃_{n \geq k} {f_{n} < a}$ , so $h_{k}$ is measurable by the sigma-algebra closure in [F1]. The sequence $h_{k}$ is increasing and its pointwise limit is ${lim inf}_{n} f_{n}$ , which is measurable since ${\lim_{k} h_{k} > a} = ⋃_{k} {h_{k} > a}$ .

2.1F1F2step 1.1∎

For every $n \geq k$ , monotonicity gives $\int h_{k} \leq \int f_{n}$ , so $\int h_{k} \leq \inf_{n \geq k} \int f_{n}$ . Taking $k \to \infty$ and applying monotone convergence to $h_{k}$ yields $\int lim inf f_{n} = \lim_{k} \int h_{k} \leq \lim_{k} \inf_{n \geq k} \int f_{n} = {lim inf}_{n} \int f_{n}$ .

Depends on

Dependency tree · 2 levels

Sources

Terence Tao, 245A Notes 3: Integration on Abstract Measure Spaces and the Convergence Theorems, full author-hosted lecture text