DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-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.

Invertible measure-preserving systems

Definition

A system in Measure-preserving transformations and systems is invertible if $T$ is a bijection and $T^{- 1} : X \to X$ is measurable. It is invertible modulo null sets if there is a measurable $X_{0} \subseteq X$ with $μ (X ∖ X_{0}) = 0$ and $T (X_{0}) = X_{0}$ , such that $T ∣_{X_{0}}$ is a bijection with measurable inverse for the trace sigma-algebra. This is an actual invariant conull restriction, not a choice of arbitrary pointwise inverses on exceptional sets.

Depends on

Measure-preserving transformations and systems

Used by

Eigenfunction for a probability system Definition
Chacon partial maps extend to an invertible map mod null sets Lemma
Koopman operators are linear isometries Theorem

Dependency tree · two levels

3 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

Einsiedler–Ward Definition 2.1 p.13 and conull restriction convention in Definition 2.7 p.16 (standard reference, not scraped)