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.
Best 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.