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arxiv: 1907.04678 · v1 · pith:JKM7WFRUnew · submitted 2019-07-09 · 🧮 math.FA

Individual ergodic theorems for infinite measure

Vladimir Chilin , Dogan Comez , Semyon Litvinov This is my paper

Pith reviewed 2026-05-25 00:02 UTC · model grok-4.3

classification 🧮 math.FA
keywords Dunford-Schwartz operatorsergodic averagesalmost uniform convergenceinfinite measure spacesBesicovitch sequencesreturn times theoremsigma-finite measures
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The pith

Dunford-Schwartz operators on L1 extend uniquely to L1 + L∞, identifying the largest subspace where ergodic averages converge almost uniformly.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that any Dunford-Schwartz operator T mapping L1 to L1 on a σ-finite infinite measure space extends uniquely to the larger space L1 + L∞. This extension is used to locate the maximal subspace R_μ inside L1 + L∞ such that the ergodic averages converge almost uniformly for every function in R_μ and every such operator T. The same almost uniform convergence is proved for averages weighted by bounded Besicovitch sequences. Invoking Assani's extension of Bourgain's return times theorem further yields pointwise convergence of those weighted averages along orbits of measure-preserving transformations, for functions in R_μ. Applications are given for fully symmetric subspaces contained in R_μ.

Core claim

Any Dunford-Schwartz operator extends uniquely to L1(Ω) + L∞(Ω), which permits the construction of the largest subspace R_μ of L1(Ω) + L∞(Ω) on which the averages (1/n) ∑ T^k(f) converge almost uniformly for every Dunford-Schwartz operator T and every f in R_μ; the same holds for weighted averages with bounded Besicovitch sequences, and pointwise convergence along orbits holds almost everywhere.

What carries the argument

The unique extension of a Dunford-Schwartz operator from L1 to the sum space L1 + L∞, which defines the maximal subspace R_μ for almost uniform convergence of ergodic averages.

If this is right

  • Ergodic averages converge almost uniformly on R_μ for every Dunford-Schwartz operator.
  • Averages weighted by any bounded Besicovitch sequence also converge almost uniformly on R_μ.
  • For any measure-preserving transformation, the weighted orbit averages converge pointwise almost everywhere for functions in R_μ.
  • The results restrict to any fully symmetric subspace contained inside R_μ.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The subspace R_μ supplies a natural setting for extending other pointwise ergodic theorems beyond finite measures.
  • Fully symmetric subspaces inside R_μ inherit the convergence properties automatically.
  • The construction may guide similar maximal subspaces for other classes of operators on infinite-measure spaces.

Load-bearing premise

The measure space must be σ-finite and infinite, and Assani's extension of Bourgain's return times theorem must hold for σ-finite measures.

What would settle it

Exhibit a Dunford-Schwartz operator on L1 that cannot be extended to L1 + L∞, or produce a function outside R_μ together with an operator T for which the ergodic averages fail to converge almost uniformly.

read the original abstract

Given a $\sigma$-finite infinite measure space $(\Omega,\mu)$, it is shown that any Dunford-Schwartz operator $T:\,\mathcal L^1(\Omega)\to\mathcal L^1(\Omega)$ can be uniquely extended to the space $\mathcal L^1(\Omega)+\mathcal L^\infty(\Omega)$. This allows to find the largest subspace $\mathcal R_\mu$ of $\mathcal L^1(\Omega)+\mathcal L^\infty(\Omega)$ such that the ergodic averages $\frac1n\sum\limits_{k=0}^{n-1}T^k(f)$ converge almost uniformly (in Egorov's sense) for every $f\in\mathcal R_\mu$ and every Dunford-Schwartz operator $T$. Utilizing this result, almost uniform convergence of the averages $\frac1n\sum\limits_{k=0}^{n-1}\beta_kT^k(f)$ for every $f\in\mathcal R_\mu$, any Dunford-Schwartz operator $T$ and any bounded Besicovitch sequence $\{\beta_k\}$ is established. Further, given a measure preserving transformation $\tau:\Omega\to\Omega$, Assani's extension of Bourgain's Return Times theorem to $\sigma$-finite measure is employed to show that for each $f\in\mathcal R_\mu$ there exists a set $\Omega_f\subset\Omega$ such that $\mu(\Omega\setminus\Omega_f)=0$ and the averages $\frac1n\sum\limits_{k=0}^{n-1}\beta_kf(\tau^k\omega)$ converge for all $\omega\in\Omega_f$ and any bounded Besicovitch sequence $\{\beta_k\}$. Applications to fully symmetric subspaces $E\subset\mathcal R_\mu$ are given.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript shows that any Dunford-Schwartz operator T : L¹(Ω) → L¹(Ω) on a σ-finite infinite measure space (Ω, μ) admits a unique extension to the sum space L¹(Ω) + L∞(Ω). This extension is then used to identify the largest subspace R_μ ⊆ L¹ + L∞ such that the ergodic averages (1/n) ∑_{k=0}^{n-1} T^k f converge almost uniformly (in the Egorov sense) for every f ∈ R_μ and every Dunford-Schwartz operator T. The paper further establishes almost-uniform convergence of weighted averages with bounded Besicovitch sequences, invokes Assani’s extension of Bourgain’s return-times theorem to obtain pointwise convergence results along orbits, and gives applications to fully symmetric subspaces of R_μ.

Significance. If the extension and maximality claims hold, the work supplies a natural maximal space for almost-uniform ergodic theorems in the infinite-measure setting, extending the classical Dunford–Schwartz theory. The Besicovitch-weighted and return-times results broaden applicability. The manuscript correctly situates its contributions relative to Assani’s prior theorem and supplies reproducible statements that can be checked against the cited external results.

minor comments (3)
  1. [Introduction / §2] The definition of the sum space L¹(Ω) + L∞(Ω) and the precise sense in which the extension is unique should be stated explicitly in the first section that introduces the operator extension (currently only alluded to in the abstract).
  2. [§1] The phrase “almost uniformly (in Egorov’s sense)” is used without a self-contained definition or reference to the precise formulation employed; a short paragraph or citation would remove ambiguity for readers unfamiliar with the infinite-measure variant.
  3. [Theorem 3.2] The statement that R_μ is the largest such subspace is central; a brief indication of how maximality is proved (e.g., by exhibiting a function outside R_μ for which some T fails to converge) would strengthen the claim even if the full argument appears later.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work and the recommendation of minor revision. No specific major comments appear in the report, so we have nothing to rebut point-by-point. We will incorporate any minor editorial suggestions that may be supplied by the editor or typesetter in the revised version.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation establishes a unique extension of Dunford-Schwartz operators from L1 to L1+L∞ on σ-finite infinite spaces and identifies the maximal subspace R_μ for a.u. convergence of ergodic averages. These steps rest on standard operator theory and invoke Assani's external extension of Bourgain's return-times theorem only for the secondary application; neither the extension nor the subspace identification reduces by the paper's own equations to a fitted parameter or self-citation chain. The central claims remain independent of the paper's internal quantities.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claims rest on standard domain assumptions of measure theory and the validity of one cited external theorem; no free parameters or new postulated entities are introduced.

axioms (3)
  • domain assumption The measure space (Ω, μ) is σ-finite and infinite.
    Explicitly stated as the setting in which all results hold.
  • domain assumption T is a Dunford-Schwartz operator from L1(Ω) to L1(Ω).
    The class of operators for which the extension and convergence statements are proved.
  • domain assumption Assani's extension of Bourgain's Return Times theorem holds for σ-finite measures.
    Invoked to obtain the pointwise convergence result for orbits.

pith-pipeline@v0.9.0 · 5845 in / 1592 out tokens · 33324 ms · 2026-05-25T00:02:55.264347+00:00 · methodology

discussion (0)

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Reference graph

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