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I'm not completely new to analysis, but I'm an engineer -- very applied, not very theoretical -- looking into self-studying pure mathematics. I've recently stumbled upon Henstock–Kurzweil integrals; many internet articles state that it's a very intuitive idea, but I simply can't wrap my head around its intuitive meaning as successfully as I can do it for the Riemann integral. The Wikipedia article seems to be pretty well-written, but I probably need a simpler stated approach/definition, because I'm only starting to get into all of these things.

Can someone give me their own explanation of what the Henstock-Kurzweil is, or perhaps a good resource?

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How about this book?… – Max Muller Aug 1 '11 at 15:04
A Garden of Integrals has a chapter on Henstock-Kurzweil, and goes into some detail about the differences with the other integrals. – Arturo Magidin Aug 1 '11 at 20:01
I'd like to second very strongly the piece of advice given by Didier Piau about Demailly's documents. Demailly's thesis is essentially that HK-integral could be taught as a first integration theory, so around the end of high school. To prove this (admittedly bold) thesis, he wrote what could be an introduction aiming this kind of audience (this is the “light” version,…) I think that's as easy as it gets. Even if your command of French is shaky, you should definitely take a look at it. – PseudoNeo Aug 1 '11 at 21:15
@PseudoNeo Thank you, though my knowledge of French is quite poor. I will definitely look into it, the diagrams alone are a great help. – Phonon Aug 2 '11 at 0:18
up vote 5 down vote accepted

There is a nice beginner's treatment in section 8.1 of Abbott's Understanding analysis.

Abbott's main source is Bartle's article "Return to the Riemann integral." JSTOR link, full article

Abbott also recommends The integral: an easy approach after Kurzweil and Henstock by Lee and Výborný and A modern theory of integration by Bartle for more detailed treatments.

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In my opinion the first and foremost place to explore should be Eric Schechter's webpage dedicated to the so-called gauge integral (linked on the WP page you mention).

One should probably also mention the book The Integral: An Easy Approach after Kurzweil and Henstock by Lee Peng Yee and Rudolf Vyborny, and, if you happen to read French, two very accessible introductory texts due to Jean-Pierre Demailly and available as [E3] on his webpage of teaching documents (their bibliography includes Yee and Vyborny's book).

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How about the Carus Monograph:
R. M. McLeod, Generalized Riemann Integral (Carus Mathematical Monographs) (Mathematical Association of America, 1982)
It may be out of print, so look in libraries.

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