Questions tagged [gauge-integral]
For questions about Henstock-Kurzweil integral or gauge integral.
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Henstock-Kurzweil integral and the Controlled Convergence Theorem
Let me recall that, for the Henstock-Kurzweil integral, there holds a very general convergence theorem:
Let $(f_n)$ be a sequence of $HK$-integrable functions on $[0, 1]$ such that
$f_n(x) \to f(x)$ ...
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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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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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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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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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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
$$\int_{a}...
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Functions that are Khinchin integrable but not Henstock-Kurzweil integrable
I have read that the Khinchin integral is more general than the Henstock-Kurzweil integral (gauge integral). I am studying gauge integration for a course right now and am not quite at the level to ...
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I need help proving a result regarding multipliers for the HK integral
A function $g:[a,b]\rightarrow \mathbb{R}$ is said to be of bounded variation on $[a,b]$ if
$$
\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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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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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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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),$ ...
user626213
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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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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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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. $$
...
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generalized Riemann Integrability of $f\cdot g$
Let $\mathcal{R}([a.b])$ the set of all Riemann-integrable functions in $[a,b]$.
Let $\mathcal{R}^{*}([a,b])$ the set of all Generalized Riemann-Integrable functions in $[a,b]$ (I'm talking about the ...
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Every partition $P$ $\delta$-fine has $c$ as tag if and only if $\delta(x)\leq |x-c|$
Let $\delta$ a gauge of $[a,b]$. How can I prove the following?
Every $\delta$-fine tagged partition $\mathcal{P}$ of $[a,b]$ has $c$ as tag $\Leftrightarrow$ $\delta(x)\leq|x-c|$ for all $x\in[a,b]-\...
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Equivalence of Lebesgue and Henstock-Kurzweil (Gauge) integral.
If $f$ is Henstock-Kurzweil integrable $\Longrightarrow$ $f$ is measurable.
$f$ is Lebesgue integrable $\Longleftrightarrow$ $|f|$ is Henstock-Kurzweil integrable.
$|f|$ is Henstock-Kurzweil ...
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Show that the following function has a gauge-integral and calculate its value.
We define:
$$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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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