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Why it's possible to integrate the function: $$f(x)=\frac{1}{x}\sin\left(\frac{1}{x^\alpha}\right)$$ using Kurzweil Henstok integral while it's not Lebesgue integrable because the singularity in $x=0$? Thanks.

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A short answer could be that Lebesgue integration is based on “R is complete as an ordered space” while Kurzweil-Henstock integration is based on “R is complete as a normed space” and the function in your example is oscillating around 0, thus exhibiting a very depleasant behaviour when considered from an “R is ordered” point of view while it is not that worse when considered from a “R is a normed space” point of view.

If you are looking for a more precise statement, the point is that the wether the positive part nor the negative part of the function in your example is integrable, so you cannot construct its Lebesgue integrale. Since Kurzweil-Henstok proceeds differently, it is not an issue in this theory.

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