arxiv: 2604.18618 · v2 · submitted 2026-04-16 · 🧮 math.GM

Recognition: 2 theorem links

· Lean Theorem

On Fubini type theorems for the Riemann integral

Ilgar Jabbarov , Jeyhun Abdullayev

Authors on Pith no claims yet

Pith reviewed 2026-05-12 00:55 UTC · model grok-4.3

classification 🧮 math.GM

keywords Riemann integralFubini theoremrepeated integrationJordan measureLebesgue measuremultidimensional integrationiterated integrals

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The pith

In important applications a modification of repeated integration makes the Riemann integral as powerful as the Lebesgue integral via Fubini's theorem.

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

The evaluation of multidimensional integrals is straightforward by repeated integration over parallelepipeds, but for general domains the Riemann integral faces complications from the distinction between Jordan and Lebesgue measures. In contrast, Fubini's theorem resolves this easily for Lebesgue integrals. This paper establishes that in certain important applications of the Riemann integral, a modified version of the repeated integration theorem can be formulated. Under this modification, the theorem operates with the same effectiveness as Fubini's theorem does for Lebesgue integrals. Readers interested in integral calculus would care because it extends the usability of Riemann integrals to more complex domains without requiring a switch to Lebesgue theory.

Core claim

One of the essential questions of the theory of multidimensional integrals concerns the evaluation of integrals taken in given domains. In the simplest case, when integrating over parallelepipeds, evaluation can easily be performed by repeated integration. In the case of the Lebesgue integral, the question is easily solvable by Fubini's theorem. In the case of the Riemann integral, the situation is complicated by the difference between Jordan and Lebesgue measures. In this paper, we show that in certain important applications of Riemann integrals, one can establish a modification of the theorem on repeated integration in which Fubini's theorem is as powerful as in the case of the Lebesgue.

What carries the argument

A modification of the theorem on repeated integration that overcomes the Jordan-Lebesgue measure difference for Riemann integrals in specific applications.

If this is right

Multidimensional Riemann integrals in those applications can be computed via repeated one-variable integrations.
The power of iteration matches that available for Lebesgue integrals.
Applications relying on Riemann integrals gain a reliable tool for domain evaluation.
Computations avoid the need to invoke Lebesgue theory for iteration purposes.

Where Pith is reading between the lines

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

Practitioners in fields using Riemann integrals, such as physics or engineering, might continue using them for complex regions.
This could prompt further research into precise conditions making the modification valid.
Similar modifications might apply to other integral properties differing between Riemann and Lebesgue.

Load-bearing premise

That there exist certain important applications where the modification of the repeated integration theorem can be established despite the general differences between Jordan and Lebesgue measures.

What would settle it

An explicit counterexample consisting of a domain and integrand in an important application where the modified repeated integration does not yield the same result or effectiveness as Fubini's theorem would for the Lebesgue integral of the same function.

Figures

Figures reproduced from arXiv: 2604.18618 by Ilgar Jabbarov, Jeyhun Abdullayev.

**Figure 1.** Figure 1: Theorems 3, 4, and 5 take precedence over Theorem 2 in applications, which is obvious from their formulation. Despite Theorem 2 establishing the Fubini theorem for the Riemann integral in its full generality, for applications, the definition of the function F(x) in that form for the repeated integral is not appropriate. But in formulations of Theorems 3 and 4, this integral has a definite meaning even thou… view at source ↗

read the original abstract

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 claims a modification of repeated integration for Riemann integrals that works like Fubini in some applications, but the abstract supplies no statement, conditions, or proof, leaving the claim uncheckable.

read the letter

The main point is that the authors want a version of the repeated-integration theorem for Riemann integrals that avoids the usual limitations from Jordan measure and gives results as strong as Fubini for Lebesgue in certain practical cases. They note the complication with domains beyond simple parallelepipeds and suggest the modification can be made to work without full Lebesgue tools. That direction makes sense if the details hold up, since many applications still prefer Riemann for its direct connection to limits and approximations. The paper does identify the right technical gap between the two theories. The soft spot is that nothing concrete appears: no explicit statement of the modified theorem, no list of the domains or functions it covers, and no sketch of why the equality survives without absolute integrability or completeness. The stress-test note is accurate on this—the claim rests on an unspecified restriction whose validity cannot be checked from what is given. If the full manuscript actually states the precise conditions and shows the result by direct Riemann arguments, it would be a modest but usable technical fix. Without that, the central assertion stays out of reach. This work is aimed at analysts who need to stay inside Riemann integration for computational or conceptual reasons rather than switch to Lebesgue. It would deserve a serious referee if the full text contains the theorem, a proof, and checks against standard counterexamples; otherwise the manuscript is too thin to review. I would recommend sending it to peer review only after confirming the full version supplies the missing statement and argument.

Referee Report

3 major / 1 minor

Summary. The paper claims that despite the distinction between Jordan and Lebesgue measures, a modification of the repeated-integration theorem for the Riemann integral can be established in certain important applications, making Fubini-type results as powerful as in the Lebesgue setting.

Significance. If the modification is made explicit with verifiable conditions and a self-contained proof that avoids Lebesgue completeness or absolute integrability, the result could allow direct use of iterated Riemann integrals over Jordan-measurable domains in applications where the full Lebesgue machinery is unnecessary, thereby simplifying some multidimensional computations.

major comments (3)

The manuscript never states the precise form of the proposed modification (e.g., the exact equality between the double Riemann integral over D and the iterated integrals, together with the hypotheses on D and f that guarantee it). Without this statement the central claim cannot be evaluated.
No proof, sketch, or verification is supplied for the asserted modification, nor is any error analysis or counter-example check provided to confirm that the equality survives under only Riemann integrability.
The class of domains and functions to which the modification applies is described only as 'certain important applications' and 'domains where the Riemann integral exists'; the paper supplies neither a concrete characterization (e.g., Jordan-measurable sets with continuous boundary) nor an example that can be checked directly.

minor comments (1)

The abstract and introduction repeat the same high-level claim without advancing to a formal statement; a dedicated theorem environment would improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough and constructive report. The comments correctly identify that the current manuscript lacks an explicit statement of the main result, a proof, and a precise characterization of the domains and functions involved. We will undertake a major revision to address each of these points directly while preserving the paper's focus on a Riemann-integral modification that avoids Lebesgue machinery.

read point-by-point responses

Referee: The manuscript never states the precise form of the proposed modification (e.g., the exact equality between the double Riemann integral over D and the iterated integrals, together with the hypotheses on D and f that guarantee it). Without this statement the central claim cannot be evaluated.

Authors: We agree that the precise statement is absent. In the revised manuscript we will insert a clearly labeled theorem that states: Let D be a bounded Jordan-measurable set in R^2 whose boundary has Jordan measure zero, and let f be Riemann integrable on D. Then the double Riemann integral of f over D equals the iterated Riemann integrals, provided the inner integrals exist for almost every fixed outer variable (in the Jordan sense). This formulation will be placed immediately after the introduction. revision: yes
Referee: No proof, sketch, or verification is supplied for the asserted modification, nor is any error analysis or counter-example check provided to confirm that the equality survives under only Riemann integrability.

Authors: The observation is accurate; the present text contains no proof. We will add a self-contained proof that proceeds from the definition of the Riemann integral via partitions and upper/lower sums, using only the Jordan measurability of D and the Riemann integrability of f. The argument will not invoke Lebesgue completeness or absolute integrability. We will also include a brief discussion of why the result fails for certain non-Jordan-measurable sets, together with a simple numerical verification on a rectangle and on a disk. revision: yes
Referee: The class of domains and functions to which the modification applies is described only as 'certain important applications' and 'domains where the Riemann integral exists'; the paper supplies neither a concrete characterization (e.g., Jordan-measurable sets with continuous boundary) nor an example that can be checked directly.

Authors: We accept that the description is too vague. The revision will replace the phrase 'certain important applications' with an explicit class: bounded Jordan-measurable domains whose boundary is a finite union of C^1 curves (hence has Jordan measure zero) and functions that are continuous on the closure of D or, more generally, Riemann integrable on D. We will also supply a fully worked example over the unit disk with a continuous integrand, showing both the double integral and the iterated integrals computed directly from Riemann sums. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central claim is a direct demonstration without reduction to inputs or self-citations.

full rationale

The abstract presents the result as showing a modification of the repeated-integration theorem for Riemann integrals in certain applications, where Fubini-type results match Lebesgue power despite Jordan-Lebesgue distinctions. No equations, derivations, or load-bearing steps are visible in the provided text that equate outputs to inputs by construction, fit parameters then rename them as predictions, or rely on self-citations for uniqueness. The claim is framed as an independent mathematical demonstration under restricted domains where Riemann integrals exist, making the derivation self-contained against external benchmarks rather than circular. Vagueness in conditions is a rigor issue, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard distinction between Jordan and Lebesgue measures plus the existence of Riemann integrals in the domains considered; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (1)

standard math Standard properties of the Riemann integral and the difference between Jordan and Lebesgue measures
Invoked implicitly when stating that the situation is complicated by this difference.

pith-pipeline@v0.9.0 · 5401 in / 1083 out tokens · 41462 ms · 2026-05-12T00:55:08.413778+00:00 · methodology

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Works this paper leans on

9 extracted references · 9 canonical work pages

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