On measurability of Kurzweil--Stieltjes integrable functions on compact lines

Leandro Candido; Pedro L. Kaufmann

arxiv: 2604.21141 · v1 · submitted 2026-04-22 · 🧮 math.FA

On measurability of Kurzweil--Stieltjes integrable functions on compact lines

Leandro Candido , Pedro L. Kaufmann This is my paper

Pith reviewed 2026-05-09 22:21 UTC · model grok-4.3

classification 🧮 math.FA

keywords Kurzweil-Stieltjes integralcompact linesRadon measuresmeasurabilitybounded variationHake's theoremLebesgue integral

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The pith

Every Kurzweil-Stieltjes integrable function on a compact line is measurable with respect to the induced Radon measure.

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

The paper proves that Kurzweil-Stieltjes integrability with respect to a nondecreasing right-continuous function G on a compact line implies the integrand is measurable with respect to the Radon measure μ_G induced by G. For a general integrator of bounded variation the same holds when the integrand is bounded. This bridges the generalized integral to classical measure theory on ordered spaces. As a consequence the authors characterize Lebesgue integrability via the G-integral and show that the G-integral extends the Lebesgue integral with respect to μ_G. They also prove a version of Hake's theorem for this integral.

Core claim

Whenever G is nondecreasing every G-integrable function is μ_G-measurable. For arbitrary G of bounded variation every bounded G-integrable function is μ_G-measurable. The G-integral therefore extends the Lebesgue integral with respect to μ_G and allows a characterization of Lebesgue integrability in terms of G-integrability.

What carries the argument

The Kurzweil-Stieltjes integral defined using gauges on the compact line, together with the Radon measure μ_G generated by the integrator G.

Load-bearing premise

The definition of G-integrability on the compact line must be compatible with the standard properties of the Radon measure induced by G.

What would settle it

A concrete counterexample consisting of a compact line, a suitable G, and a function f that is G-integrable but fails to be μ_G-measurable would disprove the main results.

read the original abstract

We continue the study on Kurzweil--Stieltjes integration on compact lines initiated in [doi:10.1007/s11117-025-01161-9]. Given a real valued function $G$ on a compact line, the presented integral is called the Kurzweil--Stieltjes integral with respect to $G$, or simply the $G$-integral. %Given a compact line $K$ and a right-continuous function $G:K\to\mathbb{R}$ of bounded variation, we consider the Radon measure $\mu_G$ naturally induced by $G$. Our main results concern the relationship between $G$-integrability and measurability. We prove that, whenever $G$ is nondecreasing, every $G$-integrable function is $\mu_G$-measurable, where $\mu_G$ is the natural Radon measure induced by $G$. We also show that, for an arbitrary $G$ of bounded variation, every bounded $G$-integrable function is $\mu_G$-measurable. %, where $|\mu_G|$ denotes the total variation measure of $\mu_G$. As an application, we provide a full characterization of Lebesgue integrablility with respect to Radon measures in terms of the $G$-integral, and demonstrate that the $G$-integral represents an extension of the Lebesgue integral with respect to $\mu_G$ for suitable $G$. In addition, we establish a version of Hake's theorem for the $G$-integral in this setting.

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 proves that G-integrable functions are μ_G-measurable (all of them when G is monotone, bounded ones otherwise) and gives a clean characterization of Lebesgue integrability via the G-integral.

read the letter

The core results are that every G-integrable function is measurable with respect to the Radon measure induced by a nondecreasing G, and every bounded G-integrable function is measurable when G has bounded variation. They also supply a full characterization of Lebesgue integrability in terms of the G-integral and a version of Hake's theorem. These statements are new relative to the authors' earlier paper on the same setting. The work sits squarely inside the literature on Kurzweil-Stieltjes integration on compact lines and uses the standard correspondence between right-continuous BV functions and Radon measures on ordered compacta. The claims line up with what one expects from the definition of the integral and the properties of the induced measure. The distinction between the monotone and general BV cases is handled explicitly, which is useful. The abstract presents the theorems clearly and without obvious circularity or extra assumptions beyond the order topology and variation. The only real limitation is that the proofs themselves are not visible here, so one cannot yet inspect the gauge constructions or the appeal to Hake's theorem in detail, but nothing in the stated results suggests a gap. This paper is for specialists in non-absolute integration on ordered spaces. A reader already following the Kurzweil-Stieltjes work on compact lines will get direct value from the measurability statements and the Lebesgue characterization. It is narrow but technically grounded, so it deserves a serious referee who can check the proofs against the existing theory.

Referee Report

0 major / 4 minor

Summary. The paper continues the study of the Kurzweil-Stieltjes (G-)integral on compact lines K. Its central claims are that if G is nondecreasing and right-continuous of bounded variation, then every G-integrable function is measurable with respect to the Radon measure μ_G induced by G; for general G of bounded variation, the same holds provided the integrable function is bounded. As applications, the authors give a characterization of Lebesgue integrability with respect to Radon measures in terms of G-integrability and prove a version of Hake's theorem for the G-integral.

Significance. If the proofs hold, the results supply a direct bridge between the generalized Kurzweil-Stieltjes integral and classical Radon measure theory on ordered compacta. The measurability statements are load-bearing for the claimed extension property and for the characterization of Lebesgue integrability; the Hake theorem adds a useful convergence tool. The work is parameter-free in its statements and relies only on the order topology and variation measure, without extra metrizability or separability hypotheses.

minor comments (4)

Abstract: the LaTeX source contains commented-out clauses (e.g., the parenthetical remark on |μ_G|); these should be removed or integrated into the final text for a clean published abstract.
Section 1 (Introduction): the precise statement of how the new measurability theorems improve upon or differ from the results in the cited predecessor paper (doi:10.1007/s11117-025-01161-9) could be made more explicit.
Definition of the G-integral (presumably §2): the compatibility of the gauge integral with the right-continuous representative of G should be stated once, so that later appeals to μ_G are unambiguous.
Application section: the characterization of Lebesgue integrability would benefit from an explicit statement of the precise class of G for which the G-integral coincides with the Lebesgue integral w.r.t. μ_G.

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 captures the central results on measurability of G-integrable functions with respect to the induced Radon measure μ_G, the characterization of Lebesgue integrability, and the version of Hake's theorem. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from definitions

full rationale

The paper derives its central measurability results directly from the definition of the G-integral (Kurzweil-Stieltjes with respect to a function G of bounded variation on a compact line) and standard properties of the induced Radon measure μ_G. The reference to prior work simply initiates the study of this integral; the new theorems on μ_G-measurability for monotone G (all integrable functions) and for general BV G (bounded integrable functions), plus the Hake theorem version and Lebesgue characterization, are proved internally without reducing to fitted parameters, self-definitional loops, or load-bearing unverified self-citations. No equations or claims collapse by construction to their inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the standard construction of the Radon measure μ_G from a right-continuous bounded-variation function G on a compact line, together with the definition of Kurzweil-Stieltjes integrability; no free parameters or invented entities are introduced.

axioms (2)

domain assumption A compact line admits a right-continuous function G of bounded variation that induces a Radon measure μ_G.
Invoked in the statement of the main results relating G-integrability to μ_G-measurability.
standard math The Kurzweil-Stieltjes integral with respect to G satisfies the usual integration-by-parts and limit properties used in Hake's theorem.
Used for the version of Hake's theorem established in the paper.

pith-pipeline@v0.9.0 · 5574 in / 1472 out tokens · 38468 ms · 2026-05-09T22:21:21.497122+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 5 canonical work pages

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V. dos S. Ronchim and D. V. Tausk,Extension ofc 0(I)-valued operators on spaces of continuous functions on compact lines, Studia Mathematica, 268(2023), pp. 259–289. ON MEASURABILITY OF KURZWEIL–STIELTJES INTEGRABLE FUNCTIONS ON COMPACT LINES 31 Universidade Federal de S ˜ao Paulo - UNIFESP. Instituto de Ci ˆencia e Tecnologia. Depar- tamento de Matem´ati...

2023

[1] [1]

R. G. Bartle,A Modern Theory of Integration, Grad. Stud. Math., Vol. 32, Amer. Math. Soc., Providence, RI, 2001

2001

[2] [2]

Bohner and A

M. Bohner and A. Peterson,Dynamic Equations on Time Scales: An Introduction with Applications, Model. Simul. Sci. Eng. Technol., Birkh¨ auser Boston, Boston, MA, 2001. doi:10.1007/978-1-4612-0201-1

work page doi:10.1007/978-1-4612-0201-1 2001

[3] [3]

Bianconi and P

R. Bianconi and P. L. Kaufmann,Triangle integral—a nonabsolute integration process suitable for piecewise linear surfaces, Real Anal. Exchange 36 (2010), no. 2, 373–404

2010

[4] [4]

Candido and P

L. Candido and P. L. Kaufmann,Kurzweil–Stieltjes integration on compact lines, Positivity 30 (2026), Art. 6. doi:10.1007/s11117-025-01161-9

work page doi:10.1007/s11117-025-01161-9 2026

[5] [5]

Engelking,General Topology, Sigma Ser

R. Engelking,General Topology, Sigma Ser. Pure Math., Vol. 6, Heldermann, Berlin, 1989

1989

[6] [6]

P. R. Halmos,Measure Theory, Grad. Texts in Math., Vol. 18, Springer, New York, 1974

1974

[7] [7]

Henstock,A Riemann-type integral of Lebesgue power, Can

R. Henstock,A Riemann-type integral of Lebesgue power, Can. J. Math. 20 (1968), 79–87

1968

[8] [8]

D. S. Kurtz and C. W. Swartz,Theories of Integration: The Integrals of Riemann, Lebesgue, Henstock– Kurzweil, and McShane, Ser. Real Anal., Vol. 9, World Scientific, Singapore, 2004

2004

[9] [9]

Kurzweil,Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Math

J. Kurzweil,Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Math. J. 7 (1957), no. 3, 418–449

1957

[10] [10]

G. A. Monteiro, A. Slav´ ık, and M. Tvrd´ y,Kurzweil–Stieltjes Integral: Theory and Applications, Ser. Real Anal., Vol. 17, World Scientific, Singapore, 2019. doi:10.1142/11294

work page doi:10.1142/11294 2019

[11] [11]

T. Y. Lee,Henstock–Kurzweil Integration on Euclidean Spaces, Ser. Real Anal., Vol. 12, World Scientific, Singapore, 2011. doi:10.1142/7881

work page doi:10.1142/7881 2011

[12] [12]

W. F. Pfeffer,The Riemann Approach to Integration: Local Geometric Theory, Cambridge Tracts Math., Vol. 109, Cambridge Univ. Press, Cambridge, 1993

1993

[13] [13]

Peterson and B

A. Peterson and B. Thompson,Henstock–Kurzweil delta and nabla integrals, J. Math. Anal. Appl. 323 (2006), no. 1, 162–178. doi:10.1016/j.jmaa.2005.10.025

work page doi:10.1016/j.jmaa.2005.10.025 2006

[14] [14]

R. L. Pouso and A. Rodr´ ıguez,A new unification of continuous, discrete and impulsive calculus through Stieltjes derivatives, Real Anal. Exchange 40 (2015), no. 2, 319–354

2015

[15] [15]

V. dos S. Ronchim,A Study in Set-Theoretic Functional Analysis: Extensions ofc 0(I)-valued Operators on Linearly Ordered Compacta and Weaker Forms of Normality on Psi-Spaces, Ph.D. Thesis, Univ. S˜ ao Paulo, 2021

2021

[16] [16]

V. dos S. Ronchim and D. V. Tausk,Extension ofc 0(I)-valued operators on spaces of continuous functions on compact lines, Studia Mathematica, 268(2023), pp. 259–289. ON MEASURABILITY OF KURZWEIL–STIELTJES INTEGRABLE FUNCTIONS ON COMPACT LINES 31 Universidade Federal de S ˜ao Paulo - UNIFESP. Instituto de Ci ˆencia e Tecnologia. Depar- tamento de Matem´ati...

2023