False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

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FALSE: Fatou's lemma is always an equality

Statement

For every sequence of nonnegative measurable functions $(f_{n})$ , $\int {lim inf}_{n} f_{n} d μ = {lim inf}_{n} \int f_{n} d μ .$

Facts & Assumptions

Given: The statement above.

[L1]

Fatou's lemma only asserts the inequality $\leq$ (Fatou's lemma).

Refutation

technique · direct

1.1

On $R$ with Lebesgue measure, let $f_{n} : = χ_{[n, n + 1]}$ ; then ${lim inf}_{n} f_{n} = 0$ pointwise.

givenconstruct

2.1

Therefore $\int {lim inf}_{n} f_{n} d μ = 0$ , while $\int f_{n} d μ = 1$ for every $n$ . So the equality in the Statement fails, and [L1] is strict here.

step 1.1L1algebra∎

Depends on

Fatou's lemma

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

Gerald B. Folland, Real Analysis, 2nd ed., Fatou's Lemma 2.18 (standard reference, not scraped)