On $L^p$-spaces of functions with values in locally convex spaces

Humberto Prado; Matthieu F. Pinaud

arxiv: 2605.30191 · v2 · pith:43X6PX77new · submitted 2026-05-28 · 🧮 math.FA

On L^p-spaces of functions with values in locally convex spaces

Matthieu F. Pinaud , Humberto Prado This is my paper

Pith reviewed 2026-06-30 11:05 UTC · model grok-4.3

classification 🧮 math.FA

keywords Lusin measurabilitylocally convex spacesL^p spacessimple functionsdyadic approximationsvector-valued functionsnon-metrizable spaces

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The pith

Lusin-measurable functions valued in locally convex spaces admit density of simple functions in the associated L^p spaces.

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

The paper defines Lusin measurability for functions taking values in locally convex spaces, including non-metrizable ones. It studies the behavior of pointwise limits of such sequences and notes certain pathological effects in the non-metrizable case. The central results establish that simple functions are dense in the constructed L^p spaces whenever the target space is Hausdorff and that dyadic approximations produce convergence in these spaces.

Core claim

Lusin measurability extends to functions with values in locally convex spaces in a way that yields well-behaved L^p spaces, with simple functions dense for Hausdorff targets and convergence obtained through dyadic approximations.

What carries the argument

Lusin measurability for locally convex space valued functions, which supports the construction of L^p spaces with the stated approximation properties.

If this is right

Simple functions are dense in L^p for any Hausdorff locally convex target.
Dyadic approximations converge in the L^p norm for these spaces.
Pointwise limits of Lusin-measurable sequences remain Lusin-measurable when the target is metrizable.
Pathological limit behaviors appear once the target space fails to be metrizable.

Where Pith is reading between the lines

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

The construction may allow integration of functions valued in non-normable spaces without requiring metrizability.
Similar density statements could be tested for other notions of measurability in the same setting.
The results open the possibility of extending classical L^p theory to targets that are only locally convex and Hausdorff.

Load-bearing premise

Lusin measurability can be defined for functions valued in possibly non-metrizable locally convex spaces so that the resulting L^p spaces have dense simple functions and admit dyadic convergence.

What would settle it

An explicit Hausdorff locally convex space together with a Lusin-measurable function whose L^p distance to every simple function remains bounded away from zero.

read the original abstract

We study Lusin-measurable functions with values in locally convex spaces. In particular, the behavior of pointwise limits of sequences of Lusin-measurable functions and exhibit pathological phenomena arising in the nonmetrizable setting. Moreover, we establish approximation and density results for $L^p$-spaces constructed with this notion of measurability, including the density of simple functions in Hausdorff locally convex spaces and convergence results obtained through dyadic approximations.

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 defines a version of Lusin measurability for functions into locally convex spaces and proves density of simple functions plus dyadic approximation in the resulting L^p spaces, with some notes on pathologies in the non-metrizable case.

read the letter

The core of this paper is a definition of Lusin measurability that works for vector-valued functions into Hausdorff locally convex spaces, possibly non-metrizable, followed by proofs that simple functions are dense in the associated L^p spaces and that dyadic approximations give convergence results. It also flags some bad behavior of pointwise limits when the target space is not metrizable.

What stands out is the explicit handling of the non-metrizable setting and the concrete density statements; most classical treatments stop at Banach or Fréchet targets, so this fills a small but real gap in the vector-valued integration literature. The claims about simple-function density and dyadic approximation look like the sort of technical result that specialists might actually use.

The main limitation is that the abstract gives no hint of the precise definition or the proof strategy, so it is impossible to check whether the measurability notion is strong enough to avoid the usual counterexamples or whether the density holds without extra topological assumptions. If the full proofs are clean and the definition is natural, the paper is solid; if the definition turns out to be ad-hoc, the results shrink in value. No circularity or invented objects are visible from what is stated.

This is narrow functional-analysis work aimed at people already reading papers on vector measures and Bochner-type integrals. A reader outside that subfield will not find much to take away. It is worth sending to a referee who knows the Lusin-measurability literature, because the topic is legitimate and the stated results are not obviously false or redundant.

Referee Report

0 major / 3 minor

Summary. The paper defines a notion of Lusin measurability for functions taking values in (possibly non-metrizable) locally convex spaces. It analyzes the stability of this class under pointwise limits and exhibits pathological counterexamples that arise when metrizability fails. It then constructs the associated L^p spaces and proves that simple functions are dense when the target space is Hausdorff, together with convergence theorems obtained via dyadic approximations.

Significance. If the stated density and approximation results hold, the work supplies a usable integration theory for vector-valued functions in the general Hausdorff locally convex setting, which is relevant for applications involving weak topologies, inductive limits, or distribution spaces. Explicit treatment of the non-metrizable pathologies is a positive feature. The paper supplies concrete constructions rather than abstract existence statements.

minor comments (3)

[Abstract and Theorem 4.3] The abstract claims density of simple functions 'in Hausdorff locally convex spaces' but does not state whether the underlying measure space is assumed σ-finite; this hypothesis appears in the L^p construction and should be listed explicitly in the statement of the main density theorem.
[§2 and §4] Notation for the seminorms generating the locally convex topology is introduced in §2 but reused without re-statement in the definition of the L^p seminorms; a short reminder paragraph would improve readability.
[§5, paragraph following Definition 5.1] The dyadic approximation argument in §5 relies on a countable dense subset of the target space; it is not clear whether this subset is required to be dense in the original topology or only in a weaker topology, and a clarifying sentence would remove ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript and the recommendation of minor revision. The referee's description accurately reflects the paper's focus on Lusin measurability, stability under limits, pathologies in the non-metrizable case, and the density/approximation results in the Hausdorff setting.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained against external benchmarks

full rationale

The provided abstract and reader's summary describe a definition of Lusin measurability for functions valued in (possibly non-metrizable) locally convex spaces, followed by proofs of approximation, density of simple functions, and dyadic convergence results in the resulting L^p spaces. No equations, fitted parameters, or self-citations are exhibited that reduce any claimed prediction or uniqueness result to the inputs by construction. The central claims rest on the new definition and standard functional-analytic arguments rather than renaming known results or importing unverified self-citations as load-bearing. This is the normal case of an independent construction whose validity can be checked externally against the stated topological hypotheses.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no information on free parameters, axioms, or invented entities.

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Works this paper leans on

12 extracted references · 4 canonical work pages · 3 internal anchors

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work page internal anchor Pith review Pith/arXiv arXiv 1904
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Pinaud M.F.,Manifolds of absolutely continuous functions with values in an infinite-dimensional manifold and regularity properties of half-Lie groups, preprint, arXiv:2409.06512

work page internal anchor Pith review Pith/arXiv arXiv
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Tao. T,An Introduction to Measure TheoryAmerican Mathematical Society. Pinaud, Matthieu F; Departamento de Matem´ atica y Ciencia de la Computaci´ on, Universi- dad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile; matthieu.pinaud@usach.cl Prado, Humberto; Departamento de Matem´ atica y Ciencia de la Computaci´ on, Universidad de Santiago de C...

[1] [1]

Mayoral, and P

Florencio, M., F. Mayoral, and P. J. Pa´ ul,Spaces of vector-valued integrable functions and localization of bounded subsets, Math. Nachr. 174 (1995), 89–111

1995

[2] [2]

and Weis L.Analysis in Banach Spaces, Volume I: Martingales and Littlewood-Paley Theory, Ergebnisse der Mathematik und ihrer Gren- zgebiete 63, Springer, 2016

Hyt¨ onen T., Neerven J., Veraar M. and Weis L.Analysis in Banach Spaces, Volume I: Martingales and Littlewood-Paley Theory, Ergebnisse der Mathematik und ihrer Gren- zgebiete 63, Springer, 2016

2016

[3] [3]

Rodr´ ıguez,On the Completeness ofLλ for Locally Convex Spaces, Arch

Jim´ enez P., B. Rodr´ ıguez,On the Completeness ofLλ for Locally Convex Spaces, Arch. Math. (Basel) 52 (1989), 82-91

1989

[4] [4]

Gl¨ ockner, H.,Measurable regularity properties of infinite-dimensional Lie groups, preprint, arXiv:1601.02568

work page internal anchor Pith review Pith/arXiv arXiv

[5] [5]

Gl¨ ockner, H.,Lie Groups of Measurable Mappings, Canad. J. Math. 55 (2003), 969-999

2003

[6] [6]

Gl¨ ockner, H., K-H Neeb,Infinite-Dimensional Lie Groups, preprint, arXiv:2602.12362

work page arXiv

[7] [7]

Nikitin, N.,Measurable regularity of infinite-dimensional Lie groups based on Lusin mea- surability, preprint, arXiv:1904.10928

work page internal anchor Pith review Pith/arXiv arXiv 1904

[8] [8]

Pinaud M.F.,Manifolds of absolutely continuous functions with values in an infinite-dimensional manifold and regularity properties of half-Lie groups, preprint, arXiv:2409.06512

work page internal anchor Pith review Pith/arXiv arXiv

[9] [9]

Rodr´ ıguez B. and S. Palero,On the uniform limit of quasi-continuous functions, Rev. R. Acad.Cien. Serie A. Mat. VOL. 95 (1), 2001, pp. 29-37

2001

[10] [10]

Rudin,W.,Real and Complex Analysis, McGraw-Hill, 3nd ed., 1987

1987

[11] [11]

Stone A.A.,Topology and measure, Measure theory, U3-U8, Proc. Conf. Oberwolfach, 1975- Lecture Notes in Mathematics, 541. Springer-Verlag, Berlin, Heidelberg, New York, 1976

1975

[12] [12]

T,An Introduction to Measure TheoryAmerican Mathematical Society

Tao. T,An Introduction to Measure TheoryAmerican Mathematical Society. Pinaud, Matthieu F; Departamento de Matem´ atica y Ciencia de la Computaci´ on, Universi- dad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile; matthieu.pinaud@usach.cl Prado, Humberto; Departamento de Matem´ atica y Ciencia de la Computaci´ on, Universidad de Santiago de C...