# Questions tagged [gauge-integral]

For questions about Henstock-Kurzweil integral or gauge integral.

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### Choosing a gauge for to prove a function has a Henstock-Kurzweil integral

This is a problem that I had already stumbled upon Riemann integrals anc choosing partitions. Given a function and asked to prove that is Henstock-Kurzweil integral by definition, how do you find the ...
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### Questions regarding Gauge and Darboux Integrals?

Can the Gauge Integral exist on a function defined on a countable set? What would it equal? I was wondering because if we took the Darboux Integral of $f:C\cap[a,b]\to\mathbb{R}$ where $C$ is a ...
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### Show that the following function has a gauge-integral and calculate its value.

We define: $$f: I = [0,1] \rightarrow \mathbb{R}: x \mapsto f(x) = \sin(x)1_{[0,1] - \mathbb{Q}} + x1_{[0,1]\cap\mathbb{Q}}$$ where $1_A$ is the characteristic function on A. Show that $f$ has a ...
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### Conditions for applying the second fundamental theorem of calculus with gauge integrals

I was thinking about this question while walking home today and can't seem to prove or come up with a counterexample myself. Let $f:[a,b]\rightarrow\mathbb{R}$ be a continuous function, $f(x),$ ...
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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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### Are bound functions always Henstock-kurzweil integrable?

Is there any function $f:[\alpha,\beta]\rightarrow\mathbb{R}$ that is bound but not Henstock-kurzweil integrable? I assume such a function would have to be horrendously discontinuous but I am unable ...
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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 ...
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### Henstock-Kurzweil integral of sin(x)/x by directly using explicit gauge.

I am trying to show that the Henstock-Kurzweil integral of sin(x)/x from zero to infinity is $\pi/2$. In order to do this, I wish to construct an explicit gauge and use Cousin's lemma to find a Perron ...
329 views

### Can someone give an example of a function that is not Henstock-Kurzweil/gauge integrable?

I am looking for an example of a function that is not Henstock-Kurzweil integrable. Can anybody help me?
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### Definition of Henstock integral function over a set

I understand the definition of Henstock integrable function on $[a, b]$, i.e., $f$ is Henstock-Kurzweil integrable on $[a, b]$ if there is $A \in \mathbb{R}$ with property for every $\varepsilon>0$...
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### generalized Riemann Integrability of $f\cdot g$

Let $\mathcal{R}([a.b])$ the set of all Riemann-integrable functions in $[a,b]$. Let $\mathcal{R}^{*}([a,b])$ the set of all Generalized Riemann-Integrable functions in $[a,b]$ (I'm talking about the ...
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### Is the derivative of $x \cos\left(\frac{\pi}{x}\right)$ integrable?

Let $F:[0,1]\rightarrow \mathbb{R}$ defined by $$F(x)= \left\{ \begin{array} .x\cdot \cos\left(\frac{\pi}{x}\right), &\textrm{if } x\in[0,1] \\ 0, &\textrm{ if x=0} \end{array} \right.$$ ...
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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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### 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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### 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 ...
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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 ...
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### What if we replace step functions by a different class of functions in the definition of Riemann integral?

Suppose we are working with some sets $\mathcal S_{1,2}$ of functions $[a,b]\to\mathbb R$. (Probably most often we would take $\mathcal S_1=\mathcal S_2=\mathcal S$.) Let us assume that we have some ...
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### Is there a treatment/development of the Stokes' Theorem using differential forms and the Henstock-Kurzweil integral i.e. the gauge integral?

I'm working through an analysis text independently to prepare for grad school, and the author has discussed the limitations of both the Riemann and Lebesgue integrals and only hinted at the power of ...
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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 ...
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### Henstock-Kurzweil integral of $f(x)=n$ for $x=1/n$ (and zero otherwise)

I need to prove that the function $$f(x) = \begin{cases} n & x=1/n \\ 0 & \text{ otherwise} \\ \end{cases}$$ defined on $[0,1]$ is Henstock-Kurzweil integrable. I've tried to ...
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### Step function approximation with Henstock–Kurzweil integral.

In the following I am working with the Henstock–Kurzweil integral. I would like to prove the following: Given a function $f : \mathbb{R} \rightarrow \mathbb{C}$ integrable on $[a..b]$, we have for ...
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### Generalized Riemann Integral

Is there any usage of studying the Henstock-Kurzweil integral as such ? It doesn't seem to be as popular a method of integration as the Lebesgue integral or even the Riemann-Stieltjes Integral,...
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 ...