TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

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Fatou's lemma

Statement

Let $(f_{n})$ be nonnegative measurable functions. 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: A sequence $(f_{n})$ of nonnegative measurable functions.

[L1]

Countable infima of measurable extended-real-valued functions are measurable, and monotone pointwise suprema are measurable (Closure properties of measurable functions used by the integral).

[L2]

Monotone convergence holds for the nonnegative integral (Monotone convergence for the integral).

[L3]

The nonnegative integral is monotone (Monotonicity and nonnegative homogeneity of the nonnegative integral).

Proof

technique · direct

1.1

For each $n$ , define [L1, given, construct] $g_{n} : = \inf_{k \geq n} f_{k} .$ Then each $g_{n}$ is measurable by [L1], one has $g_{n} \leq g_{n + 1}$ and $g_{n} ↑ {lim inf}_{n} f_{n}$ pointwise. Also $g_{n} \leq f_{n}$ for every $n$ .

2.1

By [L2], [step 1.1, L2, L3] ∎ $\int {lim inf}_{n} f_{n} d μ = \lim_{n} \int g_{n} d μ .$ Since $g_{n} \leq f_{n}$ , [L3] gives $\int g_{n} d μ \leq \int f_{n} d μ$ for every $n$ . Taking the limit in $n$ yields the claimed inequality.

Depends on

Used by

Reverse Fatou's lemma under an integrable majorant Corollary
Fatou can be strict and domination can fail simultaneously Counterexample
Mass can escape to infinity under pointwise convergence Counterexample
FALSE: Fatou's lemma is always an equality False statement
Separate holomorphy forces local boundedness on smaller polydiscs Lemma

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Richard F. Bass, Real Analysis for Graduate Students, Theorem 7.8 (standard reference, not scraped)
John K. Hunter, Measure Theory Notes, Theorem 4.22 (standard reference, not scraped)