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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 Yuan…
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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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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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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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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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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Gauge integral on infinite-dimensional Banach space and differentiability

Call $f:I\to F$ gauge integrable where $I = [a, b]$ is a compact interval and $F$ is a Banach space, if the usual definition holds like if $F = \mathbb{R}$, just replace absolute value by norm. How can one prove the following? Theorem 1. If $f:I\to…
Jakobian
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