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arxiv: 1907.03825 · v1 · pith:6H22T34Lnew · submitted 2019-07-08 · 🧮 math.FA

Fubini Type Theorems for the strong McShane and strong Henstock-Kurzweil integrals

Sokol Bush Kaliaj This is my paper

Pith reviewed 2026-05-25 00:37 UTC · model grok-4.3

classification 🧮 math.FA
keywords Fubini theoremstrong McShane integralstrong Henstock-Kurzweil integralBanach space valued functionsproduct intervaliterated integrals
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The pith

Fubini theorems hold for the strong McShane and strong Henstock-Kurzweil integrals of Banach space valued functions on rectangles in the plane.

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

The paper proves that for functions taking values in a Banach space and integrable in the strong McShane sense or the strong Henstock-Kurzweil sense over a product interval [a1,b1] times [a2,b2], the double integral equals the corresponding iterated integrals. A sympathetic reader would care because this reduces computation of the two-dimensional integral to successive one-dimensional integrations, just as the classical Fubini theorem does for Lebesgue integrals. The results apply only when the relevant integrals exist on the full product domain and on the slices.

Core claim

For a Banach space valued function that is strong McShane integrable or strong Henstock-Kurzweil integrable on the closed rectangle [a,b] subset R^2, the double integral equals the iterated integral obtained by first integrating with respect to one variable and then the other, whenever the one-dimensional integrals exist.

What carries the argument

Fubini-type theorems that equate the strong McShane (respectively strong Henstock-Kurzweil) double integral over the product interval to the iterated integrals formed from the corresponding one-dimensional strong integrals.

If this is right

  • The double integral can be evaluated by successive one-variable integrations for both the strong McShane and strong Henstock-Kurzweil cases.
  • The order of integration can be reversed when both iterated integrals exist.
  • These equalities hold without requiring absolute integrability of the function.

Where Pith is reading between the lines

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

  • The same reduction technique may apply to higher-dimensional rectangles if analogous strong integrals are defined there.
  • Applications that rely on iterated integration, such as solving integral equations, could now use these non-absolute integrals in two variables.

Load-bearing premise

The functions must belong to the classes in which the strong McShane or strong Henstock-Kurzweil integrals exist on the product interval.

What would settle it

A concrete Banach-valued function on a rectangle that is strong McShane integrable over the whole domain yet yields a double integral unequal to either iterated integral would falsify the claim.

read the original abstract

In this paper, we will prove Fubini type theorems for the strong McShane and strong Henstock-Kurzweil integrals of Banach spaces valued functions defined on a closed non-degenerate interval $[a,b] =[a_{1}, b_{1}] \times [a_{2}, b_{2}] \subset \mathbb{R}^{2}$.

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

0 major / 1 minor

Summary. The paper claims to prove Fubini-type theorems for the strong McShane and strong Henstock-Kurzweil integrals of Banach space-valued functions on the product rectangle [a,b]=[a1,b1]×[a2,b2]⊂R², under the standing assumption that the double integral exists.

Significance. If the proofs hold, the results would extend Fubini theorems to these non-absolute vector-valued integrals, which could be of interest in the theory of Henstock-Kurzweil and McShane integration in Banach spaces.

minor comments (1)
  1. The provided abstract states the claim but contains no derivation outline, lemmas, or key steps, making it impossible to assess the technical content or soundness from the given information alone.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for reviewing our manuscript on Fubini-type theorems for the strong McShane and strong Henstock-Kurzweil integrals of Banach space-valued functions. The report notes the standing assumption that the double integral exists and gives an uncertain recommendation, but lists no specific major comments. We maintain that the proofs in the paper are complete and correctly handle the stated assumptions for the product interval in R².

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained under stated assumptions

full rationale

The paper's central task is to prove Fubini-type theorems for the strong McShane and strong Henstock-Kurzweil integrals of Banach-valued functions on a product rectangle, under the explicit hypothesis that the double integral exists on the product interval. This is a standard conditional proof: the integrability assumption is granted as a premise, after which the paper must show that the iterated integrals exist and equal the double integral. No equations, fitted parameters, or self-citations are exhibited that reduce the claimed result to its own inputs by construction. The derivation therefore remains independent of the target statement and does not trigger any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work is a pure existence proof in real analysis; no numerical parameters are fitted and no new entities are postulated.

axioms (1)
  • standard math Standard properties of Banach spaces, closed intervals, and the definitions of the strong McShane and strong Henstock-Kurzweil integrals.
    These are background facts from functional analysis and integration theory invoked to state the setting.

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discussion (0)

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Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

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