Questions tagged [gauge-integral]

For questions about Henstock-Kurzweil integral or gauge integral.

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79
votes
5answers
7k 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 ...
8
votes
2answers
620 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 ...
12
votes
1answer
885 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 ...
4
votes
1answer
299 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,...
3
votes
1answer
328 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?
1
vote
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 ...
6
votes
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 ...