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arxiv: 2507.16762 · v1 · submitted 2025-07-22 · 🧮 math.FA

Almost uniform vs. pointwise convergence from a linear point of view

L. Bernal-Gonz\'alez , M.C. Calder\'on-Moreno , P.J. Gerlach-Mena , J.A. Prado-Bassas This is my paper

Pith reviewed 2026-05-19 03:03 UTC · model grok-4.3

classification 🧮 math.FA
keywords measurable functionspointwise convergencealmost uniform convergencevector subspacesalgebrassequencesconvergence modes
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The pith

Under natural assumptions on the measure space, large vector subspaces and algebras exist in the families of sequences of measurable functions that converge to zero pointwise almost everywhere but not almost uniformly, and in those that do,

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

The paper reviews comparisons between different modes of convergence for sequences of measurable functions and focuses on the algebraic structures of the families involved. It proves that under natural assumptions there exist large vector subspaces as well as large algebras inside the family of sequences converging pointwise almost everywhere to zero but failing to converge almost uniformly. The same holds for the family of sequences that converge almost uniformly but not uniformly almost everywhere. A sympathetic reader cares because these results give a linear and algebraic perspective on how the gaps between convergence modes can be filled with many independent or multiplicatively closed examples.

Core claim

The authors prove the existence of large vector subspaces as well as large algebras contained in the family of the sequences of measurable functions converging to zero pointwise almost everywhere but not almost uniformly, and in the family of the sequences of measurable functions converging to zero almost uniformly but not uniformly almost everywhere.

What carries the argument

Large vector subspaces and algebras constructed inside the families of sequences distinguished by their specific convergence behaviors.

If this is right

  • The family of sequences converging pointwise almost everywhere but not almost uniformly is closed under linear combinations within large subspaces.
  • The corresponding family for almost uniform but not uniform convergence likewise admits large algebras.
  • These algebraic objects complement earlier results that compared the same modes of convergence without the linear structure.
  • The constructions work for sequences of measurable functions on the assumed measure spaces.

Where Pith is reading between the lines

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

  • The same linear-algebraic approach could be tested on other pairs of convergence modes to see whether large subspaces appear there too.
  • Results of this kind suggest a way to measure the 'size' of convergence classes by the dimension of subspaces they contain.
  • It would be natural to check whether the constructions extend when the measure space is not sigma-finite.

Load-bearing premise

The underlying measure space satisfies natural assumptions such as sigma-finiteness that make the constructions of the subspaces and algebras possible.

What would settle it

A concrete sigma-finite measure space in which one of the two families contains no infinite-dimensional vector subspace would falsify the existence result.

Figures

Figures reproduced from arXiv: 2507.16762 by J.A. Prado-Bassas, L. Bernal-Gonz\'alez, M.C. Calder\'on-Moreno, P.J. Gerlach-Mena.

Figure 1
Figure 1. Figure 1: First iterations of the typewriter sequence. the interval [0, 1] (see [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
read the original abstract

A review of the state of the art of the comparison between any two different modes of convergence of sequences of measurable functions is carried out with focus on the algebraic structure of the families under analysis. As a complement of the amount of results obtained by several authors, it is proved, among other assertions and under natural assumptions, the existence of large vector subspaces as well as of large algebras contained in the family of the sequences of measurable functions converging to zero pointwise almost everywhere but not almost uniformly, and in the family of the sequences of measurable functions converging to zero almost uniformly but not uniformly almost everywhere.

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

1 major / 2 minor

Summary. The manuscript reviews comparisons between modes of convergence (pointwise a.e., almost uniform, uniformly a.e.) for sequences of measurable functions, with emphasis on algebraic structure. It proves, under natural assumptions on the measure space, the existence of large vector subspaces and large algebras inside the family of sequences converging to zero pointwise a.e. but not almost uniformly, and inside the family converging almost uniformly but not uniformly a.e.

Significance. The results complement prior work on convergence modes by exhibiting rich algebraic structure in the 'difference sets' between convergence classes. Explicit constructions of subspaces and algebras (when fully detailed) constitute a concrete strength, as they provide falsifiable, checkable examples rather than abstract existence.

major comments (1)
  1. [Abstract and §1 (setting)] The precise 'natural assumptions' on the underlying measure space are invoked to guarantee non-emptiness of the target families but are not listed explicitly. This is load-bearing: on finite-measure spaces Egorov's theorem forces pointwise a.e. convergence to imply almost uniform convergence, emptying the first family and rendering the subspace/algebra claims vacuous. The setting (likely §2 or the preliminaries) must state the exact hypotheses (e.g., σ-finiteness plus existence of sets of infinite measure, or non-atomicity) and verify that the constructed sequences fail almost-uniform convergence on a set of positive measure.
minor comments (2)
  1. Notation for the families (e.g., how 'large' is quantified—dimension, codimension, or algebraic independence) should be fixed at first use for clarity.
  2. A short table or diagram comparing the four convergence modes and their implications would improve readability of the state-of-the-art review.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and the helpful observation on the presentation of the setting. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract and §1 (setting)] The precise 'natural assumptions' on the underlying measure space are invoked to guarantee non-emptiness of the target families but are not listed explicitly. This is load-bearing: on finite-measure spaces Egorov's theorem forces pointwise a.e. convergence to imply almost uniform convergence, emptying the first family and rendering the subspace/algebra claims vacuous. The setting (likely §2 or the preliminaries) must state the exact hypotheses (e.g., σ-finiteness plus existence of sets of infinite measure, or non-atomicity) and verify that the constructed sequences fail almost-uniform convergence on a set of positive measure.

    Authors: We agree that the natural assumptions on the measure space must be stated explicitly in order to make the non-emptiness of the families and the validity of the subspace and algebra constructions fully transparent. In the revised manuscript we will add, at the beginning of Section 2, a precise statement of the hypotheses (σ-finiteness together with the existence of sets of infinite measure) and we will include a short verification that each constructed sequence fails almost-uniform convergence on a set of positive measure. revision: yes

Circularity Check

0 steps flagged

Existence proofs for subspaces and algebras rely on independent constructions under stated assumptions

full rationale

The paper reviews prior comparisons of convergence modes for measurable functions and then establishes new existence results for large vector subspaces and algebras inside the indicated families. These results are obtained via direct constructions that invoke standard measure-space hypotheses (such as sigma-finiteness) without reducing any target family or algebraic structure to a self-definition, a fitted parameter renamed as a prediction, or a load-bearing self-citation chain. The central claims are therefore self-contained and do not collapse by construction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claims rest on standard measure-theoretic axioms and the unspecified 'natural assumptions' on the measure space; no free parameters, invented entities, or ad-hoc postulates are indicated in the abstract.

axioms (1)
  • standard math Standard properties of measurable functions on a measure space (e.g., sigma-additivity, completeness)
    Invoked implicitly to define pointwise a.e. and almost uniform convergence.

pith-pipeline@v0.9.0 · 5643 in / 1191 out tokens · 37111 ms · 2026-05-19T03:03:55.030259+00:00 · methodology

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Reference graph

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