Questions tagged [gauge-integral]

For questions about Henstock-Kurzweil integral or gauge integral.

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Henstock-Kurzweil Integrability of Product Functions

If $f$ Henstock-Kurzweil integrable on $[0,1]$, is the function $xf(x)$ also HK integrable on $[0,1]$? More generally, given a $C^\infty$ function $\phi(x)$, is the product function $f\phi$ HK ...
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35 views

Prove that if $|f|$ is gauge integrable then so is $f$.

I have tried to prove this using the Cauchy criterion for HK-integration but I have been unsuccessful thus far. I have various fancy theorems at my disposal such the MCT and the DCT but I can not ...
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26 views

Henstock integrable

I doubt if this interval is true and satisfy this $$ x_{i-1}^{2} < \xi_{i}^{2} - \dfrac{1}{3}(x_{i}^{2} + x_{i}x_{i-1} + x_{i-1}^{2}) < x_{i}^{2} < 2\delta^{2}.$$ here is example $$\int_{a}...
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57 views

Functions that are Khinchin integrable but not Henstock-Kurzweil integrable

I have read that the Khinchin integral is more general than the Henstock-Kurzweil integral (gauge integral). I am studying gauge integration for a course right now and am not quite at the level to ...
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1answer
219 views

I need help proving a result regarding multipliers for the HK integral

A function $g:[a,b]\rightarrow \mathbb{R}$ is said to be of bounded variation on $[a,b]$ if $$ \sup\left\{\sum_{i=1}^n|g(x_i)-g(x_{i-1})|:a = x_0 <\ldots<x_n = b\right\}<+\infty. $$ If $g:[...
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63 views

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 ...
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16 views

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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1answer
124 views

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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1answer
50 views

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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1answer
64 views

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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69 views

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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42 views

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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1answer
111 views

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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1answer
85 views

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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1answer
96 views

Every partition $P$ $\delta$-fine has $c$ as tag if and only if $\delta(x)\leq |x-c|$

Let $\delta$ a gauge of $[a,b]$. How can I prove the following? Every $\delta$-fine tagged partition $\mathcal{P}$ of $[a,b]$ has $c$ as tag $\Leftrightarrow$ $\delta(x)\leq|x-c|$ for all $x\in[a,b]-\...
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167 views

Equivalence of Lebesgue and Henstock-Kurzweil (Gauge) integral.

If $f$ is Henstock-Kurzweil integrable $\Longrightarrow$ $f$ is measurable. $f$ is Lebesgue integrable $\Longleftrightarrow$ $|f|$ is Henstock-Kurzweil integrable. $|f|$ is Henstock-Kurzweil ...
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76 views

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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79 views

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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98 views

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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1answer
76 views

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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403 views

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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1answer
126 views

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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1answer
66 views

Gauge Integral: Non-Borel Spaces

Is there a gauge integral over non Borel spaces? (My interest lies in finite measure spaces.) It seems as the definition of the gauge integral crucially depends on the existence of open sets for a ...
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1answer
407 views

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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1answer
384 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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1answer
176 views

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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2answers
698 views

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 ...
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1answer
343 views

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,...
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144 views

Proof of $f=0$ a.e. in $[a,b]$ then $f$ is gauge integrable and $\int_a^bf=0$

Let $f:[a,b]\to \mathbb{R}$ so that $f(x)=0$ almost everywhere in $[a,b]$. Prove that $f$ is gauge integrable and $\int_a^bf=0$. How can this be proven using the following definition of measure: $\mu(...
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2answers
623 views

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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0answers
165 views

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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1answer
953 views

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 ...
3
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3answers
723 views

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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5answers
9k views

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 ...