Questions tagged [gauge-integral]

For questions about Henstock-Kurzweil integral or gauge integral.

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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 ...
6
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0answers
58 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 ...
6
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0answers
85 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 ...
3
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0answers
76 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 ...
3
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1answer
166 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 ...
2
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0answers
33 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 $\varepsilon>0$...
2
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0answers
164 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 ...
1
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1answer
46 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, $f(x),$ ...
1
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0answers
132 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 ...
1
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1answer
69 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 ...
1
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0answers
138 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(...
0
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0answers
15 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 ...
0
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0answers
56 views

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 ...