Mathematics

Questions tagged [gauge-integral]

For questions about Henstock-Kurzweil integral or gauge integral.

Henstock-Kurzweil integral or gauge integral (sometimes also called generalized Riemann integral, Perron integral, Denjoy integral or Luzin integral) is a generalization of Riemann integral.

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Why are gauge integrals not more popular?

A recent answer reminded me of the gauge integral , which you can read about here . It seems like the gauge integral is more general than the Lebesgue integral, e.g. if a function is Lebesgue integrable, it is gauge integrable. (EDIT - as Qiaochu…
Chris Brooks
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Integrability: Neither improper Riemann nor Lebesgue but Henstock-Kurzweil

Can you think of a function that is neither improper Riemann nor Lebesgue integrable, but is Henstock-Kurzweil integrable? I'd like to put a bounty on this question, but my reputation is not nearly enough yet. Translated to math, find $f$ such that…
Samuel Tan
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A very useful lemma for Henstock-Stieltjes integration

I'd like to see a proof (or hints and outlines) for the following lemma, which is very useful to prove some interesting properties, including an Integration by Parts theorem for Henstock-Stieltjes integrals: Let $f$, $g$ and $\varphi$ be (normally…
Pedro Vaz Pimenta
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Is every "almost everywhere derivative" Henstock–Kurzweil integrable?

It is well known that the Henstock–Kurzweil integral fixes a lot of issues with trying to integrate derivatives. The second fundamental theorem of calculus for this integral states: Given that $f : [a,b] \rightarrow \mathbb{R}$ is a continuous…
Sam Forster
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Mistake in Bartle's proof of Hake's Theorem?

Here is Bartle's proof of Hake's Theorem found in "A Modern Theory of Integration". I think there is a mistake in the highlighted line: The Theorem: $f:[a,b]\to \mathbb{R}$ is gauge integrable if and only if it is gauge integrable in $[a,x]\ \forall…
Optional
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Equivalence of the Lebesgue integral and the Henstock–Kurzweil integral on nonnegative real functions

Let $f:[a,b]\to[0,\infty)$ ($\mathbb{R}\ni a<b\in\mathbb{R}$), and fix $c\geq0$. I want to establish the equivalence of the concepts of Lebesgue integrability and Henstock–Kurzweil integrability for this class of functions. In particular, consider…
triple_sec
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What makes Cousin's theorem remarkable?

I've stumbled upon a mention of Cousin's theorem in the context of Henstock–Kurzweil integral and got confused. I do not understand why this fact is called a theorem and what makes it any remarkable, when it seems to be a more or less trivial…
Alexey
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Can the LCT and MCT for Lebesgue integrable functions be viewed as a lattice completeness result?

The set of Lebesgue integrable functions form a lattice under pointwise min and max (also more generally for R, Henstock-Kurzweil integrable functions with an upper or lower bound form a lattice as well). The convergence theorems look a lot like…
saolof
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What are the necessary and sufficient conditions for a function to be Henstock–Kurzweil integrable?

I recently stumbled upon Lebesgue’s criterion for Riemann integrability. It didn't take very long until I found this result quite intuitive. I then began studying the Henstock–Kurzweil integral. Very quickly I realized that finding the necessary and…
David
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Can conditionally convergent series be interpreted as a "generalized Henstock-Kurzweil integral"?

One amazing thing about the Lebesgue integral is that is defined w.r.t. to a given measure and that there a lot of different measures making the Lebesgue integration a very general tool (consider Harmonic Analysis). The Henstock-Kurzweil integral…
Stefan Perko
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Looking for an accessible explanation of Henstock–Kurzweil (gauge) integral

I'm not completely new to analysis, but I'm an engineer -- very applied, not very theoretical -- looking into self-studying pure mathematics. I've recently stumbled upon Henstock–Kurzweil integrals; many internet articles state that it's a very…
Phonon
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Are all Henstock-Kurzweil integrable functions expressible as the sum of a Lebesgue and an improper Riemann integrable function?

This question is based on this post , where in the comments, Toby Bartels conjectures that every Henstock-Kurzweil (gauge) integrable function $f\in\mathcal{HK}$ can be expressed as $f= g + h$ for a Lebesgue integrable function $g\in\mathcal{L}$ and…
user1111412
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If $f(x)$ is Henstock-Kurzweil integrable on $[a,b]$, then is $f(x)\mathrm{e}^{\mathrm{i}x}$ also Henstock-Kurzweil integrable on $[a,b]$?

I was wondering about how Fourier series behaves in the setting of Henstock-Kurzweil integration. For example, the non-Lebesgue-integrable function $f(x) = \dfrac{1}{x}\mathrm{e}^{\mathrm{i}/x}$ can have a Fourier expansion since the…
Jianing Song
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Is there an intuitive way to view the definition of the McShane integral?

To my understanding the definition of the McShane integral is identical to the definition of the Henstock–Kurzweil integral with the exception that each tag does not have to be contained in the subinterval for which it is assigned to. However, the…
David
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Example of a function f that is Generalized Riemann Integrable, but its square is NOT Generalized Riemann Integrable.

I am reading a section about Generalized Riemann Integral (Kurzweil-Henstock), and there was a problem on that section to provide an example of a function $f$ on $[0,1]$ that is Generalized Riemann Integrable, but its square $f^2$ is NOT Generalized…
Pat_Ho
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