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 enough yet. Translated to math, find $f$ such that

$$ f \notin \mathscr{L,R^*} $$ but $$ f\in \mathscr{HK} $$ where $\mathscr{HK}$ denotes the set of Henstock-Kurzweil integrable functions.

  • 2
    $\begingroup$ I could be mistaken, but doesn't the first paragraph of the wiki article on the gauge integral give your desired function? $\endgroup$ – JSchlather Oct 24 '11 at 7:06
  • $\begingroup$ Good pickup! Is that integral improper Riemann integrable? I have modified the question. $\endgroup$ – Samuel Tan Oct 24 '11 at 7:17
  • $\begingroup$ 'Translated to math' as opposed to what? $\endgroup$ – Alexei Averchenko Oct 25 '11 at 6:33
  • 1
    $\begingroup$ @AlexeiAverchenko Haha...as opposed to fluffy English. I'm just having too much fun typesetting math in LaTeX, being new to this forum. I'll get over it in a few weeks. $\endgroup$ – Samuel Tan Oct 25 '11 at 6:42

This is a blatant cheat, but anyway, here goes:

Take $f(x) = \frac{\sin{x}}{x}$, which is well-known to be improperly Riemann integrable, but not Lebesgue integrable.

Take the characteristic function $g$ of $[0,1] \cap \mathbb{Q}$ which is Lebesgue integrable but not improperly Riemann integrable.

The KH-integral integrates both, hence it integrates $h(x) = f(x) + g(x)$.

Clearly, $h(x)$ cannot be either, improperly Riemann integrable or Lebesgue integrable, because this would force $g$ or $f$ to have a property it doesn't have.

  • $\begingroup$ I'm not sure what $\chi_{[0,1]\cap \mathbb{Q}}$ is. Some kind of characteristic function? But yes, that is an answer. $\endgroup$ – Samuel Tan Oct 25 '11 at 5:11
  • 1
    $\begingroup$ @Samuel: Yes, that's the intention. $\endgroup$ – t.b. Oct 25 '11 at 5:14
  • $\begingroup$ In the spirit of t.b.'s answer, here is another example: take $D$ the Dirichlet function (not improper Riemann integrable) and add it to the Dirichlet integral, $$\int_0^{\infty} D(x) + \frac{sin(x)}{x} dx$$ This belongs to $\mathscr{HK}$ though $\endgroup$ – Samuel Tan Oct 25 '11 at 5:16
  • $\begingroup$ So is it possible that $\mathcal{HK} = \mathcal{L} + \mathcal{R}^*$; that is, every Henstock-integrable function is the sum of a Lebesgue-integrable function and an improperly Riemann-integrable function? $\endgroup$ – Toby Bartels Apr 29 '17 at 6:48

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.