Talk:Henstock-Kurzweil integral

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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). --Eruionnyron 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 ;-) Eruionnyron 19:44, 11 Feb 2005 (UTC)

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 , 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)

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

[edit] 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)