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Can you think of a function that is not improper Riemann or Lebesgue integrable, but is generalized Riemann (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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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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