# Questions tagged [gauge-integral]

For questions about Henstock-Kurzweil integral or gauge integral.

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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 ...
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### Integration by substitution, monotone version for Henstock-Kurzweil integrals

Let $\mathcal{HK}(I)$ denote the Henstock-Kurzweil integrable functions on $I$. By mimicking the case for Lebesgue integral I've proven the following: Theorem $1$. Let $F$ be an indefinite integral of ...
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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 ...
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### On the set of points where a Henstock-Kurzweil integrable function fails to be Lebesgue integrable

One example of a function that is Henstock-Kurzweil integrable but not Lebesgue integrable is $f(x) = \frac{1}{x} \cos\left(\frac{1}{x^2}\right)$ on $[0, 1]$. However, $f$ only fails to be locally-...
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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, ...
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### Some questions concerning the construction of the line integral in $\mathbb{C}$

I am comparing two ways of defining the integral along a path of a function of complex domain and value. One is the one given in Conway's Functions of one Complex Variable and the other is given in ...
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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 ...
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### Is there a constructive presentation of the Henstock-Kurzweil integral?

Treating the Riemann integral in a constructive setting is easy and straightforward. Treating the closely related but much more powerful Henstock-Kurzweil integral constructively is almost easy, ...
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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 ...
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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 ...
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### Proof that if $f,g,h:[a,b]\to \mathbb{R}$ with $h\le f,g$ and $f,g,h$ are gauge integrable then so is $\min(f,g)$

I am asking for a self contained proof of this assertion: If $f,g,h:[a,b]\to \mathbb{R}$ with $h\le f,g$ and $f,g,h$ are gauge integrable then so is $\min(f,g)$. The integral in question is the ...
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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 ...
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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; ...
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