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.

Measure-preserving transformations and systems

Definition

Let $(X, A, μ)$ be a measure space. A measurable self-map $T : X \to X$ is measure preserving if $μ (T^{- 1} E) = μ (E)$ for every $E \in A$ . The quadruple $(X, A, μ, T)$ is a measure-preserving system; it is a probability system if $μ (X) = 1$ . Here $T^{- 1} E = {x : T (x) \in E}$ denotes an inverse image, whether or not $T$ is invertible. Neither completeness nor finiteness is implicit. The measure-space and measurable-map conventions are Measure spaces and A measurable function between measurable spaces.

Depends on

Used by

Invertible measure-preserving systems Definition
Strict and mod-null invariant sigma-algebras Definition
Strong and weak mixing on a probability space Definition
The Koopman operator Definition
A preserving permutation on two equal atoms Example
The maximal ergodic inequality on a probability space Lemma
Compositions, iterates and completions preserve invariance Proposition
Integral invariance under measure-preserving maps Theorem
Measure preservation can be checked on a generating pi-system Theorem

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

E–W Definition 2.1, pp.13–14 (standard reference, not scraped)