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I posed this question on Mathematics SE recently, though by the total lack of attention it has gotten, I do not anticipate an answer and bring it here.

What are some Khinchin integrable $f$ which are not Henstock-Kurzweil integrable?

I myself have just begun studying gauge integration for a course and am unable to answer this question$-$given we apply gauge integrals to deal with non-Lebesgue / non-Riemann integrable functions, I was surprised to learn there even was a more general integral, and am curious as to what non-gauge integrable functions necessitate it.

Please let me know by a comment if this question is not appropriate for MO. I will delete it accordingly.

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