ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedaudited 2026-09-14

How statement and proof provenance work

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The invariant L2 projection for a half-rotation

Example

Assume the Axiom of Choice. For $T = R_{1 / 2}$ and $f = 1_{[0, 1 / 4)}$ , the orthogonal projection onto the invariant $L^{2}$ subspace is

$P_{M} f = \frac{1}{2} 1_{[0, 1 / 4) \cup [1 / 2, 3 / 4)} .$

Both the pointwise and $L^{2}$ ergodic averages converge to this nonconstant function.

Facts & Assumptions

Given: Full choice, Lebesgue probability on the circle, $T = R_{1 / 2}$ , and $f = 1_{[0, 1 / 4)}$ .

[F1]

The half-rotation preserves Lebesgue probability (Circle rotations preserve Lebesgue measure).

[F2]

Von Neumann's theorem says $A_{n} f \to P_{M} f$ in complex $L^{2}$ (Von Neumann mean ergodic theorem in L2).

[F3]

Birkhoff gives the pointwise almost-everywhere limit for this $L^{1}$ observable (Birkhoff pointwise ergodic theorem).

[F4]

Full choice is the assumption recorded in The Axiom of Choice.

[F5]

Half-open intervals have Lebesgue measure equal to their length (A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included).

Verification

technique · direct period-two calculation and uniqueness of the norm limit

1.1

Since $T^{2}$ is the identity, $f \circ T = 1_{[1 / 2, 3 / 4)}$ and the summands in $A_{n} f$ alternate between $f$ and $f \circ T$ .

givenalgebra

2.1

Put $g = (f + f \circ T) / 2$ . For $n = 2 q$ , $A_{n} f = g$ exactly; for $n = 2 q + 1$ , $∣ A_{n} f - g ∣ \leq 1 / n$ everywhere. Therefore $A_{n} f \to g$ both pointwise and in $L^{2}$ , since the circle has measure one.

F1step 1.1algebra

2.2

The function $g$ is $\frac{1}{2} 1_{[0, 1 / 4) \cup [1 / 2, 3 / 4)}$ . The half-rotation interchanges its two support intervals, so $g \circ T = g$ ; it is nonconstant because [F5] gives positive measure to both its support and complement.

F1F5step 1.1

3.1

By [F2], the same sequence $A_{n} f$ has $L^{2}$ limit $P_{M} f$ . Uniqueness of limits in the $L^{2}$ norm and step 2.1 therefore give $P_{M} f = g$ . Step 2.1 also strengthens the pointwise almost-everywhere conclusion of [F3] to convergence at every point in this example.

F2F3step 2.1step 2.2

4.1

Full choice is used only through the projection theorem [F2], as recorded by [F4]; the period-two computation itself is explicit.

F2F4step 3.1∎

Depends on

Used by

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Sources

Charles Walkden, Ergodic Theory lecture notes (standard reference, not scraped)