CounterexampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-29

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.

A completed-product measurable set can have a nonmeasurable exceptional section

Statement refuted

Every section of a completed-product measurable function is measurable.

Counterexample

technique · direct

Let $N \subseteq R$ be non-Lebesgue-measurable and let $E : = {0} \times N \subseteq R^{2} .$ Put $f : = 1_{E}$ .

Facts & Assumptions

Given: The function $f = 1_{E}$ above.

[L1]

The set $E = {0} \times N$ is contained in a planar null set, so it becomes measurable after completing the product measure. (A nonmeasurable subset of a null line shows that the product of complete measures need not be complete)

Verification

1.1

By [L1], the indicator $f$ is measurable for the completed product measure.

2.1

The section at $0$ is $f_{0} = 1_{N}$ , whose support $N$ is not Lebesgue measurable. Hence $f_{0}$ is not measurable. So completed-product measurability gives section measurability only almost everywhere, not at every parameter.

step 1.1∎

Depends on

A nonmeasurable subset of a null line shows that the product of complete measures need not be complete

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

John K. Hunter, Measure Theory, Example 5.20 (standard reference, not scraped)