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.

Ergodicity relative to an invariant measure

Definition

A measure-preserving system is ergodic for $μ$ if each $E \in I$ has $μ (E) = 0$ or $μ (X ∖ E) = 0$ , with $I$ as in Strict and mod-null invariant sigma-algebras. For a probability system this means $μ (E) \in {0, 1}$ . The definition is relative to the invariant measure; no probability assumption is implicit in the general null/conull formulation.

Depends on

Strict and mod-null invariant sigma-algebras

Used by

Birkhoff strong law for iid coordinate shifts Corollary
A preserving identity need not be ergodic Counterexample
Birkhoff's theorem for an ergodic probability system Theorem
Chacon transformation is ergodic Theorem
Equivalent invariant-set and invariant-function criteria for ergodicity Theorem
Mixing implies weak mixing, which implies ergodicity 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

Sarig Definition 1.4 p.5 (standard reference, not scraped)