Alphabeta Math
CounterexampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10
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 preserving identity need not be ergodic

Statement refuted

The assertion “every measure-preserving probability transformation is ergodic” is false.

Facts & Assumptions

[F1]

Every strictly invariant set in an ergodic system must be null or conull Ergodicity relative to an invariant measure.

Counterexample

Given: The objects and hypotheses in the statement.

1.1

Take X={0,1} with all subsets measurable, μ(E)=∣E∣/2, and T(x)=x. The measure is countably additive because a disjoint family has at most two nonempty members. Its total mass is one. For every subset E, T−1E=E, so T is measurable and preserves probability.

given
2.1

The set E={0} is strictly invariant and has μ(E)=μ(X∖E)=1/2>0. It is neither null nor conull, contradicting the ergodicity requirement. The specified T is therefore a counterexample.

F1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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