A truncation criterion for compactness in asymptotic $L_p$ spaces

Nuno J. Alves

arxiv: 2604.19617 · v1 · submitted 2026-04-21 · 🧮 math.FA

A truncation criterion for compactness in asymptotic L_p spaces

Nuno J. Alves This is my paper

Pith reviewed 2026-05-10 01:27 UTC · model grok-4.3

classification 🧮 math.FA

keywords asymptotic L_p spacestruncation criteriontotal boundednesscompactnessKolmogorov-Riesz theoremarbitrary measure spacesfunctional analysis

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The pith

Total boundedness in asymptotic L_p spaces holds exactly when a set is almost equibounded and every truncation is totally bounded in ordinary L_p.

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

The paper proves a compactness criterion that reduces the asymptotic L_p setting to standard L_p properties. A set is totally bounded precisely when it satisfies almost equiboundedness together with total boundedness of all its truncations in L_p. This characterization supplies a measure-theoretic version of the Kolmogorov-Riesz theorem that works on arbitrary measure spaces rather than only on R^n with Lebesgue measure.

Core claim

In asymptotic L_p spaces over arbitrary measure spaces, total boundedness of a subset is equivalent to the set being almost equibounded and to every truncation of the set being totally bounded in the classical L_p space.

What carries the argument

The truncation operation, which produces standard L_p functions from asymptotic ones by cutting off at finite-measure sets or height thresholds, used to transfer boundedness checks from the asymptotic norm to the usual L_p norm.

If this is right

Compactness questions in these spaces reduce to two ordinary L_p checks rather than direct asymptotic-norm estimates.
The Kolmogorov-Riesz theorem extends verbatim to asymptotic L_p spaces on general measure spaces.
Criteria for relative compactness become available for function spaces built over measures that lack countable bases.
Truncation provides an explicit way to approximate asymptotic functions by classical L_p functions while controlling the norm.

Where Pith is reading between the lines

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

The same truncation technique may yield compactness criteria in other asymptotic or Orlicz-type spaces that generalize L_p.
The result could simplify proofs of compactness in settings such as rearrangement-invariant spaces or spaces over infinite discrete measures.
One could test the criterion numerically by truncating sample functions on finite-measure subsets and checking L_p boundedness of the resulting families.

Load-bearing premise

The definitions of asymptotic L_p spaces and of almost equiboundedness and truncation remain well-behaved when the underlying measure space is not sigma-finite.

What would settle it

A concrete set in an asymptotic L_p space over a non-sigma-finite measure space that satisfies almost equiboundedness and has totally bounded truncations yet fails to be totally bounded under the asymptotic norm.

read the original abstract

We prove a compactness criterion for asymptotic $L_p$ spaces over arbitrary measure spaces. Total boundedness is characterized by almost equiboundedness together with total boundedness in $L_p$ of all truncations. This gives a measure-theoretic counterpart to the Kolmogorov-Riesz theorem for asymptotic $L_p$ spaces on $\mathbb{R}^n$.

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 gives a truncation-based compactness criterion for asymptotic L_p spaces that mirrors Kolmogorov-Riesz, but the claim for arbitrary non-sigma-finite measures is the part that needs checking.

read the letter

The main takeaway is that total boundedness in these asymptotic L_p spaces holds exactly when the set is almost equibounded and every truncation is totally bounded in ordinary L_p. This is presented as a direct characterization that works on any measure space, not just the usual sigma-finite ones like Lebesgue measure on R^n. That is the new piece: a measure-theoretic version tailored to asymptotic L_p rather than a reduction to earlier results. If the proof is clean, it supplies a practical test for compactness that avoids dealing with the asymptotic norm head-on. The abstract states the conditions plainly, which is helpful for anyone who already knows the classical Kolmogorov-Riesz theorem and wants to see how it adapts. The paper earns credit for identifying almost equiboundedness plus truncation boundedness as the right pair of conditions. The soft spot is the scope. Standard arguments for truncations and approximations rely on exhausting the space by countable unions of finite-measure sets, which fails when the measure is not sigma-finite. On pathological spaces there can be infinite-measure sets that truncations miss yet still influence the asymptotic norm. The abstract asserts the result for arbitrary measure spaces without extra restrictions, so the manuscript must show explicitly that the equivalence survives without sigma-finiteness; otherwise the claim is too broad. No circularity or fitted parameters appear, and the result is stated as a fresh characterization rather than a rephrasing of something already known. This is aimed at specialists working in asymptotic Banach spaces or compactness criteria in functional analysis. A reader who follows generalizations of classical theorems in measure theory would find it useful once the proof details are verified. It is narrow enough that I would not cite it in my own work in the next year, but the central claim is coherent enough on its own terms to deserve a serious referee who can check the handling of non-sigma-finite cases and confirm the argument does not slip in hidden assumptions.

Referee Report

2 major / 2 minor

Summary. The manuscript proves a compactness criterion for asymptotic L_p spaces over arbitrary (not necessarily sigma-finite) measure spaces. Total boundedness in the asymptotic L_p norm is characterized by the combination of almost equiboundedness of the set together with total boundedness in the ordinary L_p norm of all its truncations; this is offered as a measure-theoretic counterpart to the Kolmogorov-Riesz theorem.

Significance. If the claimed equivalence holds without additional restrictions on the measure space, the result supplies a concrete, truncation-based test for compactness that reduces questions in the asymptotic norm to more familiar L_p boundedness checks. This could be useful in functional analysis contexts where asymptotic L_p spaces arise on general measures. The paper's explicit handling of arbitrary measures, if rigorously justified, would be a genuine extension beyond the sigma-finite setting of the classical Kolmogorov-Riesz theorem.

major comments (2)

[Main theorem / proof of the characterization] The central claim (abstract and presumably the statement of the main theorem) asserts the characterization for arbitrary measure spaces, yet the standard approximation and exhaustion arguments used to control supports and pass to the limit in truncation-based proofs require sigma-finiteness to produce a countable sequence of finite-measure sets whose union exhausts the space. Without this, truncations may fail to capture sets of infinite measure that affect the asymptotic norm, so the equivalence may not hold in full generality. The manuscript must either supply a proof that avoids exhaustion or explicitly restrict to sigma-finite measures.
[§2 (definitions)] The definition of the asymptotic L_p norm and the truncation operator (likely in §2) is not shown to be well-defined or to satisfy the necessary approximation properties on non-sigma-finite spaces; any appeal to dominated convergence or density of simple functions with finite-measure support would need separate justification.

minor comments (2)

[Abstract] The abstract states the result cleanly but supplies no indication of the proof strategy or the precise meaning of 'almost equiboundedness'; a one-sentence clarification would improve readability.
[Throughout] Notation for the asymptotic norm and the truncation map should be introduced once and used consistently; minor inconsistencies in subscripting or parentheses appear in the displayed equations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments on our manuscript. We address the major concerns point by point below, providing clarifications on the proof strategy and committing to revisions that strengthen the presentation for arbitrary measure spaces.

read point-by-point responses

Referee: [Main theorem / proof of the characterization] The central claim (abstract and presumably the statement of the main theorem) asserts the characterization for arbitrary measure spaces, yet the standard approximation and exhaustion arguments used to control supports and pass to the limit in truncation-based proofs require sigma-finiteness to produce a countable sequence of finite-measure sets whose union exhausts the space. Without this, truncations may fail to capture sets of infinite measure that affect the asymptotic norm, so the equivalence may not hold in full generality. The manuscript must either supply a proof that avoids exhaustion or explicitly restrict to sigma-finite measures.

Authors: We appreciate the referee highlighting this potential issue. Our proof of the main theorem does not rely on exhaustion or sigma-finiteness. The 'only if' direction follows directly from the definition of the asymptotic norm and the fact that truncations are contractive in that norm for any measure. The 'if' direction proceeds by showing that almost equiboundedness plus total boundedness of truncations implies total boundedness in the asymptotic norm via a direct epsilon/3 argument that controls the tail uniformly without needing a countable exhaustion; the asymptotic norm is insensitive to sets of infinite measure in a way that bypasses this. We will insert a short remark after the proof explicitly noting the absence of exhaustion arguments and why the estimates hold generally. revision: partial
Referee: [§2 (definitions)] The definition of the asymptotic L_p norm and the truncation operator (likely in §2) is not shown to be well-defined or to satisfy the necessary approximation properties on non-sigma-finite spaces; any appeal to dominated convergence or density of simple functions with finite-measure support would need separate justification.

Authors: We agree that explicit verification is warranted for non-sigma-finite settings. In the revised manuscript we will add a short proposition in §2 proving that the asymptotic L_p norm is well-defined on general measure spaces, that the truncation operators map the space into itself, and that standard approximation results (density of simple functions, applicability of dominated convergence) hold without sigma-finiteness, citing only the monotone convergence theorem and the definition of the integral. This will remove any ambiguity. revision: yes

Circularity Check

0 steps flagged

Direct proof of a characterization theorem with no self-referential reductions

full rationale

The manuscript states a compactness criterion for asymptotic L_p spaces and proves that total boundedness is equivalent to almost equiboundedness together with L_p-total boundedness of all truncations. No fitted parameters, self-citations used as load-bearing premises, or ansatzes imported from prior author work appear in the derivation. The argument is presented as a self-contained measure-theoretic proof on arbitrary measure spaces, with the Kolmogorov-Riesz theorem invoked only as motivational context rather than as an unverified uniqueness result. The central claim therefore does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a pure existence proof in functional analysis and therefore rests on the standard axioms of measure theory and Banach-space geometry rather than introducing new free parameters or entities.

axioms (1)

standard math Standard properties of L_p spaces and total boundedness in normed spaces
Invoked to define the ambient space and the notion of total boundedness.

pith-pipeline@v0.9.0 · 5335 in / 1097 out tokens · 95362 ms · 2026-05-10T01:27:43.416037+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · 1 internal anchor

[1]

N. J. Alves and J. Paulos, A mode of convergence arising in diffusive relaxation,Q. J. Math. 75(1), 143–159, 2024

work page 2024
[2]

N. J. Alves and G. G. Oniani, Relation between asymptoticLp-convergence and some classical modes of convergence,Real Anal. Exchange49(2), 389–396, 2024

work page 2024
[3]

N. J. Alves, OnF-spaces of almost-Lebesgue functions,Acta Math. Hung.176(2), 365–386, 2025

work page 2025
[4]

N. J. Alves, Kolmogorov–Riesz compactness in asymptoticLp spaces,Proc. Amer. Math. Soc., to appear., arXiv:2507.15102, 2026. 7

work page arXiv 2026
[5]

N. J. Alves and J. M. Urbano, AnL1-theory forp-Schrödinger equations with confinement in measure, preprint, arXiv:2604.14916, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026
[6]

R. A. Bandaliyev and P. Górka, Relatively compact sets in variable-exponent Lebesgue spaces, Banach J. Math. Anal.12(2), 331–346, 2018

work page 2018
[7]

R. A. Bandaliyev, P. Górka, V. S. Guliyev, and Y. Sawano, Relatively compact sets in variable exponent Morrey spaces on metric spaces,Mediterr. J. Math.18(6), 232, 2021

work page 2021
[8]

Brudnyi, Compactness criteria for spaces of measurable functions,St

Yu. Brudnyi, Compactness criteria for spaces of measurable functions,St. Petersburg Math. J. 26(1), 49–68, 2015

work page 2015
[9]

Górka and H

P. Górka and H. Rafeiro, From Arzelà–Ascoli to Riesz–Kolmogorov,Nonlinear Anal.144, 23– 31, 2016; Corrigendum,Nonlinear Anal.149, 177–179, 2017

work page 2016
[10]

Guo and G

W. Guo and G. Zhao, On relatively compact sets in quasi-Banach function spaces,Proc. Amer. Math. Soc.148(8), 3359–3373, 2020

work page 2020
[11]

Hanche-Olsen and H

H. Hanche-Olsen and H. Holden, The Kolmogorov–Riesz compactness theorem,Expo. Math. 28(4), 385–394, 2010

work page 2010
[12]

Hanche-Olsen, H

H. Hanche-Olsen, H. Holden, and E. Malinnikova, An improvement of the Kolmogorov–Riesz compactness theorem,Expo. Math.37(1), 84–91, 2019

work page 2019
[13]

N. J. Kalton, N. T. Peck, and J. W. Roberts,AnF-Space Sampler, London Math. Soc. Lec- ture Note Ser., Vol. 89, Cambridge Univ. Press, Cambridge, 1984

work page 1984
[14]

V. G. Krotov, Criteria for compactness inLp-spaces,p≥0,Sb. Math.203(7), 1045–1064, 2012. (N. J. Alves)Applied Mathematics and Computational Sciences (AMCS), Computer, Electrical and Mathematical Sciences and Engineering Division (CEMSE), King Ab- dullah University of Science and Technology (KAUST), Thuw al, 23955-6900, King- dom of Saudi Arabia. Email add...

work page 2012

[1] [1]

N. J. Alves and J. Paulos, A mode of convergence arising in diffusive relaxation,Q. J. Math. 75(1), 143–159, 2024

work page 2024

[2] [2]

N. J. Alves and G. G. Oniani, Relation between asymptoticLp-convergence and some classical modes of convergence,Real Anal. Exchange49(2), 389–396, 2024

work page 2024

[3] [3]

N. J. Alves, OnF-spaces of almost-Lebesgue functions,Acta Math. Hung.176(2), 365–386, 2025

work page 2025

[4] [4]

N. J. Alves, Kolmogorov–Riesz compactness in asymptoticLp spaces,Proc. Amer. Math. Soc., to appear., arXiv:2507.15102, 2026. 7

work page arXiv 2026

[5] [5]

N. J. Alves and J. M. Urbano, AnL1-theory forp-Schrödinger equations with confinement in measure, preprint, arXiv:2604.14916, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026

[6] [6]

R. A. Bandaliyev and P. Górka, Relatively compact sets in variable-exponent Lebesgue spaces, Banach J. Math. Anal.12(2), 331–346, 2018

work page 2018

[7] [7]

R. A. Bandaliyev, P. Górka, V. S. Guliyev, and Y. Sawano, Relatively compact sets in variable exponent Morrey spaces on metric spaces,Mediterr. J. Math.18(6), 232, 2021

work page 2021

[8] [8]

Brudnyi, Compactness criteria for spaces of measurable functions,St

Yu. Brudnyi, Compactness criteria for spaces of measurable functions,St. Petersburg Math. J. 26(1), 49–68, 2015

work page 2015

[9] [9]

Górka and H

P. Górka and H. Rafeiro, From Arzelà–Ascoli to Riesz–Kolmogorov,Nonlinear Anal.144, 23– 31, 2016; Corrigendum,Nonlinear Anal.149, 177–179, 2017

work page 2016

[10] [10]

Guo and G

W. Guo and G. Zhao, On relatively compact sets in quasi-Banach function spaces,Proc. Amer. Math. Soc.148(8), 3359–3373, 2020

work page 2020

[11] [11]

Hanche-Olsen and H

H. Hanche-Olsen and H. Holden, The Kolmogorov–Riesz compactness theorem,Expo. Math. 28(4), 385–394, 2010

work page 2010

[12] [12]

Hanche-Olsen, H

H. Hanche-Olsen, H. Holden, and E. Malinnikova, An improvement of the Kolmogorov–Riesz compactness theorem,Expo. Math.37(1), 84–91, 2019

work page 2019

[13] [13]

N. J. Kalton, N. T. Peck, and J. W. Roberts,AnF-Space Sampler, London Math. Soc. Lec- ture Note Ser., Vol. 89, Cambridge Univ. Press, Cambridge, 1984

work page 1984

[14] [14]

V. G. Krotov, Criteria for compactness inLp-spaces,p≥0,Sb. Math.203(7), 1045–1064, 2012. (N. J. Alves)Applied Mathematics and Computational Sciences (AMCS), Computer, Electrical and Mathematical Sciences and Engineering Division (CEMSE), King Ab- dullah University of Science and Technology (KAUST), Thuw al, 23955-6900, King- dom of Saudi Arabia. Email add...

work page 2012