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

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Von Neumann mean ergodic theorem in L2

Statement

Assume the Axiom of Choice. For a measure-preserving system and $f \in L^{2} (μ; C)$ ,

$A_{n} f ⟶ P_{M} f in L^{2},$

where $M = {g : g \circ T = g μ -a.e.}$ and $P_{M}$ is the orthogonal projection onto $M$ . Neither finite measure, ergodicity, nor invertibility of $T$ is required.

Facts & Assumptions

Given: AC, a measure-preserving system, and complex $f \in L^{2}$ .

[F1]

Composition $U_{T} g = g \circ T$ is a well-defined linear isometry on complex $L^{2}$ (Ergodic averages are measurable, representative independent, and Lp contractive).

[F2]

Under AC, Cesaro averages of any linear isometry on a closed complex $L^{2}$ subspace converge in norm to the orthogonal projection onto its fixed space, even when the isometry is not surjective (Hilbert cesaro averages converge to the fixed subspace, The Axiom of Choice).

[F3]

The fixed space of $U_{T}$ is exactly $M$ by Ergodic partial sums, time averages, and the invariant L2 subspace.

Proof

technique · direct application of the Hilbert-space Cesaro lemma

1.1

By [F1], $U_{T}$ is a linear isometry of complex $L^{2}$ . Its fixed vectors are precisely the classes $g$ with $g \circ T = g$ a.e., namely $M$ by [F3].

F1F3

2.1

The operator averages in [F2] satisfy $\frac{1}{n} \sum_{k = 0}^{n - 1} U_{T}^{k} f = \frac{1}{n} \sum_{k = 0}^{n - 1} f \circ T^{k} = A_{n} f .$ Applying [F2] therefore gives $A_{n} f \to P_{M} f$ in $L^{2}$ . AC is spent exactly in the published projection/Cesaro supplier. That supplier treats a nonsurjective isometry through its closed range, so no inverse of $T$ or of $U_{T}$ on all of $L^{2}$ has been assumed.

F2step 1.1

3.1

This argument used neither the pointwise Birkhoff theorem nor any finite-measure or ergodicity hypothesis. In particular its norm conclusion alone makes no assertion about pointwise convergence of the full sequence.

step 2.1∎

Depends on

Used by

The invariant L2 projection for a half-rotation Example
Norm convergence alone does not imply pointwise convergence False statement

Dependency tree · two levels

32 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

Omri Sarig, Lecture Notes on Ergodic Theory (2023) (standard reference, not scraped)
Charles Walkden, Ergodic Theory lecture notes (standard reference, not scraped)