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.

Strong and weak mixing on a probability space

Definition

For a measure-preserving probability system as in Measure-preserving transformations and systems, put $d_{n} (A, B) = μ (A \cap T^{- n} B) - μ (A) μ (B)$ for measurable $A, B$ and $n \geq 0$ . The system is strongly mixing if $d_{n} (A, B) \to 0$ for every such pair. It is weakly mixing if $\frac{1}{N} \sum_{n = 0}^{N - 1} ∣ d_{n} (A, B) ∣ ⟶ 0 (N \to \infty)$ for every such pair. The absolute value is inside the average. In all cases $N \geq 1$ and $T^{0}$ is the identity.

Depends on

Measure-preserving transformations and systems

Used by

Weak mixing implies strong mixing False statement
Chacon tower height correlations obstruct mixing Lemma
Mixing correlations extend to L2 functions Proposition
Mixing implies weak mixing, which implies ergodicity Theorem
Mixing is checkable on a generating pi-system Theorem
Weak mixing is equivalent to absence of nonconstant eigenfunctions 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

E–W Definitions 2.32 and 2.35, pp.49–50 (standard reference, not scraped)