Questions tagged [gauge-integral]

For questions about Henstock-Kurzweil integral or gauge integral.

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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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1 vote
1 answer
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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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6 votes
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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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  • 459
10 votes
3 answers
245 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 : [...
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1 vote
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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, $[0,1] \rightarrow \mathbb{R}$, ...
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1 vote
1 answer
47 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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2 votes
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30 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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3 votes
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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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4 votes
1 answer
234 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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3 votes
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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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  • 692
0 votes
1 answer
137 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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1 vote
1 answer
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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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1 vote
1 answer
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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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6 votes
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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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1 vote
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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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2 votes
1 answer
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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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3 votes
2 answers
122 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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2 votes
1 answer
158 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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1 vote
0 answers
233 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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1 vote
2 answers
111 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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  • 787
3 votes
0 answers
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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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6 votes
0 answers
124 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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1 vote
1 answer
98 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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11 votes
0 answers
451 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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2 votes
1 answer
218 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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3 votes
1 answer
77 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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4 votes
1 answer
487 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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  • 239
3 votes
1 answer
493 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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3 votes
1 answer
207 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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8 votes
3 answers
997 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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4 votes
1 answer
394 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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  • 307
1 vote
0 answers
154 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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  • 623
9 votes
2 answers
721 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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  • 623
2 votes
0 answers
179 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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  • 12.6k
12 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 ...
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3 votes
3 answers
846 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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  • 3,608
90 votes
5 answers
10k 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 ...
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