# 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 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.

• I could be mistaken, but doesn't the first paragraph of the wiki article on the gauge integral give your desired function? – JSchlather Oct 24 '11 at 7:06
• Good pickup! Is that integral improper Riemann integrable? I have modified the question. – Samuel Tan Oct 24 '11 at 7:17
• 'Translated to math' as opposed to what? – Alexei Averchenko Oct 25 '11 at 6:33
• @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. – Samuel Tan Oct 25 '11 at 6:42

## 1 Answer

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.

• I'm not sure what $\chi_{[0,1]\cap \mathbb{Q}}$ is. Some kind of characteristic function? But yes, that is an answer. – Samuel Tan Oct 25 '11 at 5:11
• @Samuel: Yes, that's the intention. – t.b. Oct 25 '11 at 5:14
• 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 – Samuel Tan Oct 25 '11 at 5:16
• 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? – Toby Bartels Apr 29 '17 at 6:48