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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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    @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.2011-10-25

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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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    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?2017-04-29