False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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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.
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FALSE: almost-everywhere convergence implies convergence in measure on every measure space

Statement refuted

almost-everywhere convergence implies convergence in measure on every measure space.

Facts & Assumptions

Given: Lebesgue measure on $R$ and the sequence $f_{n} : = χ_{[n, n + 1]}$ .

[L1]

Almost-everywhere convergence means pointwise convergence off a measurable null set. (Convergence almost everywhere relative to a measure)

[L2]

Convergence in measure means that for every real $ε > 0$ , $μ ({∣ f_{n} - f ∣ > ε}) \to 0$ . (Convergence in measure)

Refutation

technique · direct

1.1givenL1

Fix $x \in R$ . If $n > x + 1$ , then $x \notin [n, n + 1]$ , so $f_{n} (x) = 0$ . Thus $f_{n} (x) \to 0$ for every $x \in R$ , hence almost everywhere by [L1].

1.2givenL2

For every $n$ one has ${∣ f_{n} - 0 ∣ > 1 / 2} = [n, n + 1]$ , whose Lebesgue measure is $1$ . So the bad-set measures do not tend to $0$ , and [L2] fails.

2.1step 1.1step 1.2∎

This single sequence satisfies the premise and violates the conclusion, so the claim is false.

Depends on

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., Section 2.4, Example (ii) (standard reference, not scraped)
Terence Tao, 245A Notes 4: Modes of convergence, Example 4 (standard reference, not scraped)