TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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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.
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Mixing implies weak mixing, which implies ergodicity

Statement

Every strongly mixing probability system is weakly mixing, and every weakly mixing probability system is ergodic.

Facts & Assumptions

[F1]

Strong mixing is convergence of set correlations; weak mixing is convergence of their absolute Cesaro averages Strong and weak mixing on a probability space.

[F2]

In a probability system ergodicity means invariant sets have mass zero or one Ergodicity relative to an invariant measure.

Proof

Given: The objects and hypotheses in the statement.

1.1

For a fixed measurable pair let $d_{n} = μ (A \cap T^{- n} B) - μ (A) μ (B)$ . Strong mixing says $d_{n} \to 0$ . Given $ε > 0$ , choose m so that $∣ d_{n} ∣ < ε$ for $n \geq m$ . For $N > m$ , $N^{- 1} \sum_{n < N} ∣ d_{n} ∣ \leq N^{- 1} \sum_{n < m} ∣ d_{n} ∣ + ε$ . The first term tends to zero because it is a fixed finite sum divided by N. As $ε$ is arbitrary, weak mixing follows.

2.1

For a strictly invariant E, $T^{- n} E = E$ for every n. Thus its weak-mixing average against itself is exactly $∣ μ (E) - μ (E)^{2} ∣$ . A constant sequence tends to zero only if that constant is zero. Since $0 \leq μ (E) \leq 1$ , this gives $μ (E) = 0$ or $1$ , which is ergodicity.

F1F2∎

Depends on

Used by

Nothing in the library uses this result yet.

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 §2.7 pp.49–50; Sarig Proposition 1.2 (standard reference, not scraped)