Talk:Henstock–Kurzweil integral

Educational use
This was in the text of the article:

''This topic is somewhat esoteric: most mathematical departments do not teach it even for graduate students, and specialists in this field are numbered. Consider reading first Riemann integral (the oldest and simplest definition) or Lebesgue integral (the most common).''

I believe Uffish added that remark.

I disagree with this Point Of View. The Denjoy-Perron-Henstock-Kurzweil integral is part of the first year course in Mathematics and also Physics at the Université catholique de Louvain, Louvain-la-Neuve, Belgium. The course is given by Jean Mawhin (J. Mawhin, Analyse : fondements, techniques, évolution. De Boeck Université, Bruxelles, 1992. ISBN 2-8041-1670-0). -- 15:46, 9 Feb 2005 (UTC)


 * Well, that still doesn't make it taught at most departments. In the past proponents of this approach have written things here designed to 'promote' this approach. Not surprisingly there was a negative reaction. Charles Matthews 16:35, 9 Feb 2005 (UTC)


 * Thanks for the explanation. It sometimes seems to me that mathematicians in any field are numbered and that all of mathematics is viewed as esoteric ;-) 19:44, 11 Feb 2005 (UTC)

Relation to Lebesgue
It would be nice to see how this definition relates to Lebesgue integration beyond the statement that it "in some situations is more useful than the Lebesgue integral." It is easy to see that all Riemann integrable functions are Henstock-Kurzweil integrable, but does Henstock-Kurzweil itegrability imply Lebesgue integrability, or vice versa? Althai 05:17, 31 January 2007 (UTC)


 * If I recall correctly, at the external site, it is stated that Lebesgue integrability implies gauge integrability. Indeed, the example at the start of the article gives a function that is not Lebesgue integrable but is gauge integrable. If somebody confirms this, then maybe further emphasis can be added to article on these points DRLB 15:08, 31 January 2007 (UTC)


 * Yes, Lebesgue integrability implies gauge integrability. Lebesgue integral is equivalent to so-called McShane integral, which is a weaker form of the gauge integral. Its weakness is in removing the condition $$t_i\in[u_{i-1},u_i]$$, that is, the only thing that connects the point and its segment is the gauge. McShane integral is obviously weaker than the gauge integral, but equivalence of McShane and Lebesgue integrals takes some efforts to proof.
 * P.S. McShane and Kurzweil-Henstock integrals (together with Riemann integral) are a part of the first year calculus course by professor T.P.Lukashenko in the Moscow State University - i've been learning it ;)
 * --a_dergachev 17:48, 17 June 2007 (UTC)

Diagram
Probably a diagram similar to this should be added to the page. --128.163.161.42 (talk) 10:31, 23 March 2008 (UTC)

A few changes
I added a section called "Properties" which describes a few key properties of the gauge integral. (Also I removed the paragraph from the text about the fundamental theorem of calculus and added corresponding material in this section.) In particular, I added the characterization of gauge integrable functions in terms of Lebesgue integrable functions.

It would be nice to be more explicit about the fact that the gauge integral, unlike the Riemann and Lebesgue integrals, is a "nonasbsolute integral" i.e., integrability of f does not imply integrability of |f|.

Is it now time to remove the extremely vague and unhelpful statement about the gauge integral being "in some situations more useful" than the Lebesgue integral? We have now called attention to several particular uses: (i) integration of all derivatives; (ii) inclusion of improper Riemann integrals; (iii) arguably more elementary definition. 72.152.92.55 (talk) 23:11, 23 November 2007 (UTC)Plclark


 * Just added a statement on Hake's theorem. --a_dergachev (talk) 09:12, 14 February 2008 (UTC)

distributional Denjoy integral
In Real Analysis Exchange 33(1) pp51-82, Erik Talvila has defined an extraordinarily simple integral which takes Dejoy, Henstock, Lebesgue & Reimann as special cases. The definition is: given distribution $$f$$, if there exists a continuous function F with real limit at infinity such that $$F' = f$$ (distributional derivative) then define $$\int_{-\infty}^\infty f = F(\infty)-F(-\infty)$$. Integrable distributions form a Banach space, and its dual is $$\mathcal{BV}$$. He then derives int by parts, a very general change of variables, convergence theorems, Holder inequality and other properties.

Just giving heads up! --Pwsnafu (talk) 03:58, 2 July 2008 (UTC)

This comes quite many years late, but thanks for this heads up! The relevant paper is "The Distributional Denjoy Integral", Erik Talvila, Real Analysis Exchange 33(1) pp. 51-82. --Kaba3 (talk) 14:07, 1 December 2013 (UTC)

And, whoever is interested in that paper, is probably also interested in the papers "The Regulated Primitive Integral" and "The Lp Primitive Integral" from the same author. --Kaba3 (talk) 16:02, 2 December 2013 (UTC)

Question of cause/meaning
I question the sentence:" Due to these two important mathematicians, it is now commonly known as the Henstock–Kurzweil integral." I suspect that it is a poor translation..(?). As is, this sentence clearly implies that both Henstock and Kurzweil caused the integral to be named after themselves. Its common for mathmeticians to claim a bit too much credit, but in this case I wonder if the text is correct. I suspect that the author meant "Due to the importance of their work with/on it, it is commonly known as the Henstock-Kurzweil integral." If they did not claim priority and not lobby for a name change, would someone change the above to make clear what is being said? Thanks173.189.73.72 (talk) 23:10, 25 January 2014 (UTC)


 * Heh. Good one! Someone corrected it; it now says "Due to these two important contributions..." 67.198.37.16 (talk) 20:12, 20 September 2020 (UTC)

To use or not to use it when teaching
I use the idea of the gauge integral as a bridge from Riemann to Lebesgue integration in Sect.1 "Introduction" of my course Functions of a real variable. There you can also find my explanation, why use it as just a bridge, not instead of Lebesgue integral. Boris Tsirelson (talk) 16:11, 11 March 2016 (UTC)

... calculating that integral gives the same result by all three formulations
This seems trivially false to me. The Riemann integral of sin(x) from 0 to 2 pi is 0, while the Lebesgue integral is 4, and as far as I can understand from the definition given here, the gauge integral should be zero. I have however only had formal education in Riemann integrals and sums. Possibly f was meant to be nonnegative on [a, b]. Edit March 20: The weird definition of Lebesgue integration got me, the article is fine. Yodo9000 (talk) 11:02, 14 March 2022 (UTC)