CounterexampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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.

The translates of the unit interval converge almost everywhere to zero but not in measure

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)

Counterexample

technique · direct

1.1

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

givenL1

1.2

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.

givenL2

2.1

This sequence satisfies the premise of the refuted statement and violates its conclusion.

step 1.1step 1.2∎

Depends on

Used by

The translated unit intervals show that Egorov needs finite total measure Counterexample

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)