TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passverified 2026-09-23 (gpt-6-sol)

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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

1.1L1givenconstruct

For each $n$ , define $g_{n} : = \inf_{k \geq n} f_{k}$ . 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 $n$ .

2.1step 1.1L2L3∎

By [L2] and step 1.1, $\int {lim inf}_{n} f_{n} d μ = \lim_{n} \int g_{n} d μ$ . For every fixed $n$ and all $k \geq n$ , one has $g_{n} \leq g_{k} \leq f_{k}$ . By [L3], $\int g_{n} d μ \leq \inf_{k \geq n} \int f_{k} d μ$ . Taking the supremum over $n$ and using monotone convergence on the left gives $\int {lim inf}_{n} f_{n} d μ \leq {lim inf}_{n} \int f_{n} d μ$ , including infinite values.

Fatou's lemma

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources