TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Conditional sampled and continuous-time averages

Statement

Let $(M, ω, H, Φ, m_{0})$ be a Hamiltonian phase datum and let $R$ be a complete invariant Borel domain carrying a Borel probability $μ$ such that $μ (Φ_{s}^{- 1} B) = μ (B)$ for every $s \in R$ and every Borel $B \subseteq R$ . Fix a sampling interval $Δ > 0$ and define the time map $T = Φ_{Δ} ∣_{R}$ . Assume this particular map is ergodic for $μ$ , meaning that every Borel $E \subseteq R$ with $T^{- 1} E = E$ has $μ (E) \in {0, 1}$ . For every real $f \in L^{1} (μ)$ , the sampled averages $\frac{1}{N} \sum_{j = 0}^{N - 1} f (T^{j} z)$ converge $μ$ -almost surely and in $L^{1} (μ)$ to $\int_{R} f d μ$ . If $f$ is additionally bounded and Borel, then the continuous-time averages $\frac{1}{τ} \int_{0}^{τ} f (Φ_{s} z) d s$ converge $μ$ -almost surely and in $L^{1} (μ)$ to the same mean as $τ \to \infty$ . No ergodicity of every time map or generic Hamiltonian flow is asserted.

Facts & Assumptions

Given: A complete invariant domain, a probability $μ$ invariant under the Hamiltonian flow, $Δ > 0$ , ergodicity of $T = Φ_{Δ}$ , and the integrability or boundedness condition on $f$ in the relevant clause.

[A1]

${A C}_{ω}$ is carried for the completed-product Fubini supplier (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

Hamiltonian phase, complete-domain flow and reference measure: $Φ_{t}$ is a smooth local Hamiltonian flow with the usual flow composition law wherever defined.

[F2]

Measure-preserving transformations and systems and Ergodicity relative to an invariant measure: the selected map $T$ preserves $μ$ and is ergodic relative to $μ$ ; the cited convention tests strictly invariant sets, $T^{- 1} E = E$ .

[F3]

Birkhoff's theorem for an ergodic probability system: for a real $L^{1}$ observable and an ergodic probability-preserving map, its sampled averages converge almost surely and in $L^{1}$ to the space mean. The actual supplier statement permits noninvertible maps.

[F4]

Tonelli and Fubini for the completed product, with only almost-everywhere section measurability: for the completed product of time Lebesgue measure and $μ$ , the bounded jointly measurable integrand may be integrated in either order.

[F5]

Dominated convergence: a uniformly bounded sequence on the probability space that converges almost everywhere also converges in $L^{1}$ .

[F6]

Integral invariance under measure-preserving maps: flow invariance of $μ$ preserves the integral of an integrable observable under each $Φ_{s}$ .

Proof

Given: [A1], [F1]–[F6], and the hypotheses in the statement.

1.1A1F1F2F3given

The map $T = Φ_{Δ} ∣_{R}$ is a measurable self-map preserving $μ$ , and it is ergodic by hypothesis. Apply [F3] to the real observable $f \in L^{1} (μ)$ . This proves the sampled almost-sure and $L^{1}$ convergence to $\int_{R} f d μ$ .

1.2A1F4F6given

Now assume $f$ is bounded and Borel. Joint measurability of the flow makes $(s, z) \mapsto f (Φ_{s} z)$ measurable. Define $g (z) = Δ^{- 1} \int_{0}^{Δ} f (Φ_{s} z) d s$ . By [F4], $g$ is measurable; $∣ g ∣ \leq ∥ f ∥_{\infty}$ ; and by [F6] and Fubini, $\int_{R} g d μ = Δ^{- 1} \int_{0}^{Δ} \int_{R} f (Φ_{s} z) d μ (z) d s = \int_{R} f d μ$ .

2.1F1F3F5step 1.2

The discrete average of $g$ is exactly $N^{- 1} \sum_{j < N} g (Φ_{j Δ} z) = (N Δ)^{- 1} \int_{0}^{N Δ} f (Φ_{s} z) d s$ , by the flow composition law and the partition of the time interval into $N$ blocks of length $Δ$ . Applying the almost-sure part of [F3] to bounded $g$ gives convergence of these block averages to $\int_{R} f d μ$ . Since they are bounded by $∥ f ∥_{\infty}$ , [F5] also gives $L^{1}$ convergence along $τ = N Δ$ .

3.1step 2.1algebra∎

For arbitrary $τ = N Δ + r$ with $0 \leq r < Δ$ , let $I_{s} (z) = \int_{0}^{s} f (Φ_{u} z) d u$ . Then $∣ I_{τ} - I_{N Δ} ∣ \leq r ∥ f ∥_{\infty}$ and $∣ I_{N Δ} ∣ \leq N Δ ∥ f ∥_{\infty}$ . Hence $∣ \frac{I_{τ}}{τ} - \frac{I_{N Δ}}{N Δ} ∣ \leq \frac{2 Δ ∥ f ∥_{\infty}}{τ} .$ This deterministic bound tends to zero uniformly in $z$ , so the almost-sure and $L^{1}$ limits in step 2.1 hold for all real $τ \to \infty$ .

Remarks

The hypothesis is about the chosen sampled map $Φ_{Δ}$ . A continuous flow may be ergodic while a resonant sampled time map is not, so the theorem does not replace the explicit ergodicity assumption.

Conditional sampled and continuous-time averages

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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