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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.

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    $\begingroup$ 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, 2011 at 7:06
  • $\begingroup$ Good pickup! Is that integral improper Riemann integrable? I have modified the question. $\endgroup$
    – Samuel Tan
    Oct 24, 2011 at 7:17
  • $\begingroup$ 'Translated to math' as opposed to what? $\endgroup$
    – Alexei Averchenko
    Oct 25, 2011 at 6:33
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    $\begingroup$ @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, 2011 at 6:42

1 Answer 1

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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.

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  • $\begingroup$ 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, 2011 at 5:11
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    $\begingroup$ @Samuel: Yes, that's the intention. $\endgroup$
    – t.b.
    Oct 25, 2011 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, 2011 at 5:16
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    $\begingroup$ 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, 2017 at 6:48

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