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I'm studying the theory of integration due to Henstock-Kurzweil and read somewhere that a function is Lebesgue integrable if and only if is absolutely Henstock-Kurzweil integrable (both the function and its absolute value are integrable). I didn't find a convincing proof of this statement yet. I would appreciate if someone could clarify this for me.

pedro
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    theorem 5.52 of the textbook Theories of Integration: The Integrals of Riemann, Lebesgue, Henstock-Kurzweil, and McShane (second edition) of Douglas S. Kurtz and Charles W. Swartz – Masacroso Nov 11 '19 at 13:51
  • @Masacroso Thanks for the answer, Masacroso. The proof you mentioned has as one of the hypothesis that the function is non-negative. Is there any reason for that? (and where this assumption is used in the proof?) – pedro Nov 11 '19 at 14:07
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    of course: non-negative is equivalent to the condition of absolute integrability. I didn't read the book, sorry, I just found a link to the statement for this book – Masacroso Nov 11 '19 at 14:12
  • @Masacroso I understand. My point is that the argument in the proof could be applied to general functions, since the convergence theorems used in the proof don't require $f$ to be non-negative. This would show that the integrals are equivalent, which is not true. I'm trying to figure out where is the mistake. – pedro Nov 11 '19 at 14:19
  • Don't worry, the reference you gave helped a lot. I'll try understanding it. Thanks! – pedro Nov 11 '19 at 14:54
  • the key is in proposition 4.66: f is absolutely HK-integrable if and only if f+ and f− are HK-integrables. However it seems that there are some f HK-integrables (not absolutely) functions such that f+ is not HK-integrable. This is why we need non-negativity in theorem 4.91, read the last part when it says "by linearity f+ and f− are HK-integrable", the author is talking about proposition 4.66. However the "proof" of proposition 4.66 is unclear also :/ – Masacroso Nov 11 '19 at 16:21
  • my confusion comes from the terminology: absolutely HK-integrable means that $\int |f|$ and $\int f$ both exists (not just that $\int |f|$ exists). In Lebesgue integration the existence of $\int |f|$ imply that $\int f$ also exists because the measurability of $f$ ensures this, however in the HK-integral there isn't a concept of "measurability", as far as I know, and so it seems that there are cases where $\int|f|$ exists but $\int f$ doesnt exists – Masacroso Nov 11 '19 at 16:33

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