Skip to main content

Questions tagged [gauge-integral]

For questions about Henstock-Kurzweil integral or gauge integral.

Filter by
Sorted by
Tagged with
0 votes
0 answers
26 views

Is there a Hilbert space of Henstock–Kurzweil square-integrable integrable functions?

As is well-known, the space of square-integrable functions (say, on $[0,\,1]$) where the integral is a Riemann integral is not complete. If one completes it, one obtains the $L^{2}([0,\,1])$ Hilbert ...
linguisticturn's user avatar
4 votes
1 answer
83 views

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 ...
Jakobian's user avatar
  • 10.5k
4 votes
0 answers
38 views

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 ...
Jakobian's user avatar
  • 10.5k
2 votes
1 answer
54 views

Is the set of singularities of a gauge integral always a null set?

Let $\mathcal{R}^*(I)$ be the set of gauge integrable (generalized Riemann integrable) functions $f:I\to \mathbb{R}$ where $I = [a, b]$. Say that a point $c\in [a, b]$ is a singular for $f\in \mathcal{...
Jakobian's user avatar
  • 10.5k
1 vote
1 answer
95 views

Bartle's Modern theory of integration, theorem 9.1

Bartle's book, Modern theory of integration, focuses on Henstock-Kurzweil integrals, where $\mathcal{R}^*(I)$ denotes the set of Henstock-Kurzweil integrable functions on $I = [a, b]$. Also, $\mathcal{...
Jakobian's user avatar
  • 10.5k
5 votes
1 answer
174 views

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 ...
Peder's user avatar
  • 128
2 votes
0 answers
50 views

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-...
perplexed's user avatar
  • 336
8 votes
1 answer
295 views

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, ...
Alexey's user avatar
  • 2,210
1 vote
0 answers
57 views

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 ...
Victor's user avatar
  • 617
5 votes
1 answer
179 views

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 ...
Jianing Song's user avatar
  • 1,923
2 votes
1 answer
248 views

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, ...
saolof's user avatar
  • 649
6 votes
0 answers
59 views

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 ...
saolof's user avatar
  • 649
11 votes
3 answers
560 views

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 : [...
Sam Forster's user avatar
  • 1,392
1 vote
0 answers
77 views

Does the law of large numbers hold with the Henstock–Kurzweil integral?

If I am understanding, the law of large numbers correctly, one implication is this: Let $\lambda$ be the Lebesgue measure on $[0,1]$. Let $f_1,f_2,\dots,$ be functions, $[0,1] \rightarrow \mathbb{R}$, ...
user253846's user avatar
1 vote
1 answer
60 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 ...
David's user avatar
  • 852
2 votes
0 answers
32 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}...
notorious's user avatar
3 votes
0 answers
229 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 ...
Descartes Before the Horse's user avatar
4 votes
1 answer
240 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:[...
David's user avatar
  • 852
4 votes
0 answers
190 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 ...
David's user avatar
  • 852
0 votes
1 answer
158 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 ...
Arbuja's user avatar
  • 1
1 vote
1 answer
95 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, $f(x),$ ...
user avatar
1 vote
1 answer
117 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 ...
David's user avatar
  • 852
6 votes
0 answers
101 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 ...
David's user avatar
  • 852
1 vote
0 answers
63 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 $\varepsilon>0$...
hidup's user avatar
  • 91
2 votes
1 answer
128 views

Is the derivative of $x \cos\left(\frac{\pi}{x}\right)$ integrable?

Let $F:[0,1]\rightarrow \mathbb{R}$ defined by $$F(x)= \left\{ \begin{array} .x\cdot \cos\left(\frac{\pi}{x}\right), &\textrm{if } x\in[0,1] \\ 0, &\textrm{ if $x=0$} \end{array} \right. $$ ...
Mateus Rocha's user avatar
  • 2,666
3 votes
2 answers
220 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 ...
Mateus Rocha's user avatar
  • 2,666
2 votes
1 answer
208 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]-\...
Mateus Rocha's user avatar
  • 2,666
1 vote
0 answers
278 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 ...
miyagi_do's user avatar
  • 1,777
1 vote
2 answers
190 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 ...
Digitalis's user avatar
  • 797
3 votes
0 answers
101 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 ...
Martin Sleziak's user avatar
6 votes
0 answers
141 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 ...
Stefan Perko's user avatar
  • 12.5k
1 vote
1 answer
115 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 ...
Mark Wassell's user avatar
12 votes
0 answers
505 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 ...
Pedro Vaz Pimenta's user avatar
2 votes
1 answer
431 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 ...
CasaBonita's user avatar
3 votes
1 answer
79 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 ...
C-star-W-star's user avatar
4 votes
1 answer
535 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 ...
Pat_Ho's user avatar
  • 249
3 votes
2 answers
681 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?
Elke's user avatar
  • 31
3 votes
1 answer
219 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 ...
Shirly Geffen's user avatar
8 votes
3 answers
1k 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 ...
triple_sec's user avatar
  • 23.6k
4 votes
1 answer
433 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,...
Raghav's user avatar
  • 317
1 vote
0 answers
159 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(...
Optional's user avatar
  • 633
9 votes
3 answers
896 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 ...
Optional's user avatar
  • 633
2 votes
0 answers
185 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 ...
Nameless's user avatar
  • 13.5k
15 votes
1 answer
1k 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 ...
Samuel Tan's user avatar
4 votes
3 answers
985 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; ...
Phonon's user avatar
  • 4,058
101 votes
5 answers
12k 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 ...
Chris Brooks's user avatar
  • 7,504