Alphabeta Math
CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27
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 continuous preimage of a Lebesgue measurable set can be nonmeasurable

Statement refuted

That a continuous preimage of a Lebesgue measurable subset of R must be Lebesgue measurable.

Facts & Assumptions

Given: The Axiom of Choice, a Lebesgue measurable set E⊆R, a subset X⊆R, and a continuous map g:X→R such that g−1[E] is not Lebesgue measurable.

[L1]

Assuming the Axiom of Choice, such data exist. (A continuous preimage of a Lebesgue measurable subset of R can be nonmeasurable)

Counterexample

technique · direct
1.1

By [L1], the set E is Lebesgue measurable and the map g is continuous. [L1]

L1
2.1

The same fact [L1] says that g−1[E] is not Lebesgue measurable, which [step 1.1, L1] is exactly the required failure.

step 1.1L1∎

Depends on

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