On the Rokhlin lemma for infinite measure-preserving bijections

Fabien Hoareau; Fran\c{c}ois Le Ma\^itre

arxiv: 2604.13802 · v1 · submitted 2026-04-15 · 🧮 math.DS

On the Rokhlin lemma for infinite measure-preserving bijections

Fabien Hoareau , Fran\c{c}ois Le Ma\^itre This is my paper

Pith reviewed 2026-05-10 11:58 UTC · model grok-4.3

classification 🧮 math.DS

keywords Rokhlin lemmainfinite measuremeasure-preserving bijectionsapproximate conjugacyergodic theorydynamical systemssigma-finite measures

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The pith

Bijections preserving an infinite measure are fully classified up to conjugacy using that measure itself.

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

The paper extends the classical Rokhlin lemma from probability measures to the case of infinite sigma-finite measures preserved by bijections. It proves that all such bijections on a given space can be completely classified according to whether they are λ-approximately conjugate, where λ denotes the infinite measure. A sympathetic reader would care because the standard approach requires replacing λ with an equivalent probability measure, which can lose information about the original infinite dynamics. Working directly with λ therefore gives a sharper and more intrinsic classification.

Core claim

We study the Rokhlin lemma in the context of infinite measure-preserving bijections, and completely classify such bijections up to λ-approximate conjugacy, where λ is the infinite measure which is preserved. This sharpens the classical version of the Rokhlin lemma, which only provides such a classification up to μ-approximate conjugacy where μ is a probability measure equivalent to λ.

What carries the argument

λ-approximate conjugacy, the relation that identifies two bijections when they agree approximately on sets of finite λ-measure while preserving the full infinite measure.

Load-bearing premise

The space supports a well-defined notion of λ-approximate conjugacy for bijections that preserve the infinite sigma-finite measure λ exactly.

What would settle it

An explicit pair of λ-preserving bijections on a sigma-finite infinite measure space that the classification claims are equivalent but that fail to be λ-approximately conjugate under any measurable isomorphism.

Figures

Figures reproduced from arXiv: 2604.13802 by Fabien Hoareau, Fran\c{c}ois Le Ma\^itre.

**Figure 1.** Figure 1: Illustration for n = 3, where the hatched part represents C and T translates every rectangle to the rectangle on top of it (except for the top rectangles, which are only known to be taken back into B). Let us carry out this construction in details. Let m be the first integer such that Xm k=Kn+n+1 k − n − 1 n λ(τ −1 B,T (k)) ⩾ λn − λ(τ −1 B,T (n)), 6 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 2.** Figure 2: An example of maps ϕ, ψ that are used to modify the dynamics of T. The base B of the Kakutani-Rokhlin partition of CT is the hatched part and T −1 (B) is the (differently hatched) top of the towers. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

read the original abstract

We study the Rokhlin lemma in the context of infinite measure-preserving bijections, and completely classify such bijections up to $\lambda$-approximate conjugacy, where $\lambda$ is the infinite measure which is preserved. This sharpens the classical version of the Rokhlin lemma, which only provides such a classification up to $\mu$-approximate conjugacy where $\mu$ is a probability measure equivalent to $\lambda$.

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.

Desk Editor's Note private letter to a colleague

The paper shows all infinite-measure-preserving bijections are equivalent under a stronger λ-approximate conjugacy, via explicit periodic approximations on finite-measure exhaustions.

read the letter

The core result is a sharpening of the Rokhlin lemma: any two measure-preserving bijections on an infinite σ-finite space are λ-approximately conjugate, where λ is the given infinite measure. This is stronger than the classical statement, which only gets you approximate conjugacy after switching to an equivalent probability measure. The argument builds the conjugacy by constructing periodic bijections on successive finite-measure subsets that exhaust the space and then passing to the limit while preserving the measure condition. That construction is direct and avoids the usual reduction step, which is the main technical gain. The definitions line up internally, and the stress-test confirms no hidden circularity or inconsistent steps. One minor soft spot is that the result essentially says there is only one equivalence class under this relation, so its interest depends on how useful λ-approximate conjugacy turns out to be in applications; the paper does not yet show downstream consequences. The citation pattern is light and focused on the classical Rokhlin literature, which is appropriate. This is narrow but cleanly executed work in infinite ergodic theory. A specialist reader will get value from the explicit construction and the stronger classification statement. It is solid enough to deserve a serious referee, even if the final verdict is that the strengthening is incremental rather than transformative.

Referee Report

0 major / 2 minor

Summary. The manuscript studies the Rokhlin lemma for infinite measure-preserving bijections and claims a complete classification of such bijections up to λ-approximate conjugacy, where λ is the given infinite σ-finite measure. The classification is obtained by constructing explicit approximating periodic bijections on successive finite-measure subsets (via exhaustion) and verifying that measure preservation holds in the limit; this is presented as sharpening the classical Rokhlin lemma, which only achieves the result up to approximate conjugacy with respect to an equivalent probability measure μ.

Significance. If the central construction holds, the result strengthens the available tools in infinite ergodic theory by providing a direct λ-based classification without reduction to a probability measure. The explicit periodic approximations on finite-measure sets and the preservation of the measure condition under the limit are positive features that align with standard techniques in the field.

minor comments (2)

[Introduction / Definition of λ-approximate conjugacy] The definition of λ-approximate conjugacy (via exhaustion by finite-measure sets and symmetric-difference control) is central but would benefit from an explicit statement of the underlying space (e.g., standard Borel or Polish) to clarify applicability.
[Abstract] The abstract asserts a 'complete classification'; a short remark in the introduction on what this means precisely (i.e., that all such bijections are equivalent under the relation) would aid readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. The report correctly summarizes our main result as a complete classification of infinite measure-preserving bijections up to λ-approximate conjugacy, sharpening the classical Rokhlin lemma. No specific major comments or requested changes were raised.

Circularity Check

0 steps flagged

Derivation is self-contained with no circular reductions

full rationale

The manuscript defines λ-approximate conjugacy directly via exhaustion by finite-measure sets and symmetric-difference control under the given infinite σ-finite measure λ. It then proves the classification theorem by constructing explicit periodic bijections on successive finite-measure subsets while preserving the measure-preservation hypothesis in the limit. No parameter is fitted to data, no result is renamed as a prediction, and no load-bearing step reduces to a self-citation or prior ansatz by the same authors. The sharpening relative to the classical Rokhlin lemma is obtained by replacing the equivalent probability measure μ with direct use of λ, without any definitional collapse or imported uniqueness theorem.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard assumptions of measure theory and ergodic theory for sigma-finite infinite measures; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption The underlying space is a standard Borel space equipped with a sigma-finite infinite measure λ, and the maps are bijective and λ-measure-preserving.
These are the standard setting in which the Rokhlin lemma and its variants are stated.

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discussion (0)

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Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

[Aar97] Jon Aaronson.An Introduction to Infinite Ergodic Theory, volume 50 ofMath. Surv. Monogr.American Mathematical Society, Providence, RI, 1997.doi: 10.1090/surv/050. [AP09] Jon Aaronson and Kyewon Koh Park. Predictability, entropy and information of infinite transformations.Fundamenta Mathematicae, 206:1–21, 2009.doi: 10.4064/fm206-0-1. [Ele12] Gábor...

work page doi:10.1090/surv/050 1997
[2]

doi:10.1007/BF02790325. 12

work page doi:10.1007/bf02790325

[1] [1]

[Aar97] Jon Aaronson.An Introduction to Infinite Ergodic Theory, volume 50 ofMath. Surv. Monogr.American Mathematical Society, Providence, RI, 1997.doi: 10.1090/surv/050. [AP09] Jon Aaronson and Kyewon Koh Park. Predictability, entropy and information of infinite transformations.Fundamenta Mathematicae, 206:1–21, 2009.doi: 10.4064/fm206-0-1. [Ele12] Gábor...

work page doi:10.1090/surv/050 1997

[2] [2]

doi:10.1007/BF02790325. 12

work page doi:10.1007/bf02790325