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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 Harmonic Analysis).

The Henstock-Kurzweil integral doesn't seem anywhere nearly as general (at least not in this sense), so it looks less appealing. Although for real functions it's generally better because in that sense it is more general.

On Wikipedia it says that the Henstock–Kurzweil integral can be thought of as a "non-absolutely convergent version of Lebesgue integral" (w.r.t. to Lebesgue measure I assume).

Having just a tiny bit of hope for a more "general Henstock-Kurzweil integral" an obvious question arises:

Is there a generalization of the Henstock-Kurzweil integral that you can take w.r.t. to different "things" (similar to measures) that also includes conditionally convergent series as "integrals of sequences"?

(one should compare this idea to Lebesgue integration w.r.t. counting measure)

Martin Sleziak
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Stefan Perko
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  • search for Henstock-Kurzweil-Stieltjes integral (or alternatively, integrate functions that are constant on intervals $[n,n+1)$ with the HK integral) – user8268 Dec 27 '16 at 23:23
  • @user8268 Judging from your ( ... ) my question isn't so good, I have think about a way to fix it. – Stefan Perko Dec 27 '16 at 23:30
  • @user8268 Could you please explain what you had in mind regarding the Henstock-Kurzweil-Stieltjes integral? – Stefan Perko Dec 28 '16 at 09:52
  • just $\int f dg=\sum_n f(n)$ if $g(x)=[x]$. – user8268 Dec 28 '16 at 12:46
  • @user8268 Thanks. I tried to fix my question. You can put your comments into an answer so that I can accept it. – Stefan Perko Dec 28 '16 at 16:31
  • my comment is not an answer - I just wanted to point out something related that you might find interesting. Your question is more in the direction whether one can define a HK-like integral on other "spaces" besides $\mathbb R$ (or perhaps $\mathbb R^n$), and there I have no idea. – user8268 Dec 28 '16 at 19:24
  • @user8268 Well, you are right in thinking that this is the question I asked myself. But the question I actually asked here could be interpreted to be a bit more conversative than that. – Stefan Perko Dec 28 '16 at 19:28

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