CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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Birkhoff ergodic theorem for ergodic finite-measure systems

Statement

Assume the Axiom of Choice. Let $T$ preserve an ergodic measure $μ$ with $0 < μ (X) < \infty$ . For every real- or complex-valued $f \in L^{1} (μ)$ ,

$A_{n} f ⟶ \frac{1}{μ (X)} \int_{X} f d μ$

both $μ$ -almost everywhere and in $L^{1} (μ)$ .

Facts & Assumptions

Given: AC, the ergodic finite positive measure system, and $f$ in the Statement.

[F1]

Birkhoff supplies an invariant a.e. limit (Birkhoff pointwise ergodic theorem).

[F2]

Under AC, the finite-measure identification gives $\int_{X} f^{*} = \int_{X} f$ (Finite-measure identification of the Birkhoff limit, The Axiom of Choice).

[F3]

In an ergodic probability system every finite-valued a.e.-invariant real or complex measurable function is constant a.e. (Equivalent invariant-set and invariant-function criteria for ergodicity).

[F4]

On a finite measure space the averages converge in $L^{1}$ to their Birkhoff limit (Ergodic averages converge in Lp on finite-measure spaces).

Proof

technique · direct

1.1F1

Normalize the measure to $\hat{μ} = μ / μ (X)$ . This does not change measurable sets, null sets, invariance, or ergodicity, and $T$ preserves $\hat{μ}$ . By [F1], $A_{n} f \to f^{*}$ a.e. and $f^{*} \circ T = f^{*}$ a.e.

2.1F2F3step 1.1

Applying [F3] to the normalized probability system makes $f^{*} = c$ a.e. for some scalar $c$ . The event identity of [F2] at $E = X$ gives $c μ (X) = \int_{X} f^{*} d μ = \int_{X} f d μ,$ so $c = μ (X)^{- 1} \int_{X} f d μ$ . This is where the AC-dependent Radon–Nikodym identification is used.

3.1F4step 2.1∎

The case $p = 1$ of [F4] gives $∥ A_{n} f - f^{*} ∥_{1} \to 0$ . Combining with step 2.1 proves both modes of convergence. For complex $f$ , [F3] and the integral identity apply to its two components, producing the same complex constant formula.

Depends on

Used by

Reciprocal return frequency from Birkhoff Example
Birkhoff limit for a stationary integrable process Theorem

Dependency tree · two levels

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Sources

Alessio Del Vigna, The Birkhoff Ergodic Theorem (standard reference, not scraped)
Charles Walkden, Ergodic Theory lecture notes (standard reference, not scraped)
Omri Sarig, Lecture Notes on Ergodic Theory (2023) (standard reference, not scraped)