Talk:Lebesgue integration

message: With the addition of the new introductory material, this article takes the place of a good professor - at first giving a general indication of the questions and problems at hand, and once the ideas involved are explained a little further (with some nice examples) providing the formal framework which enables those ideas to be put into rigorous use. Great job.
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Couple of points where I can carp (I do concur with the post above: the work on this page was very worthwhile).
*The Lebesgue approach is not the most elementary area-based integration theory; that distinction goes to the Riemann integral.
Don't think that's actually true (cf. Dieudonne's Treatise on Analysis ona sub-Riemann theory).
*Uniform convergence of Fourier series.
Rare? A couple of derivatives will do.
Charles Matthews 07:24, 18 Dec 2003 (UTC)
replies: text:  Charles Matthews refers here to what Dieudonne calls regulated functions in the English translation (satisfying a condition something like: both one sided limits exist at every point = uniform norm closure of linear span of indicator functions of intervals ).  It is more elementary in the sense that all its properties follow by continuity.  Dieudonne disparages theRiemann integral (which in fact is the prevalent attitude among the mathematicians I know)
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I just removed the following text:
Correction: The improper Reimann integral does not exist for f or g since the improper Reimann Integral is defined as a double limit and you cannot subtract infinities, i.e.  the improper Reimann integral is defined as limlim∫abf(x) dx where the limits are taken as a goes to infinity and b goes to infinity.  What is true is that the Improper Cauchy Principal value (about zero) for f does in fact exist and it is PV∫-∞∞f(x)=0.
I will modify the text to say something like (sometimes called the Improper Cauchy Principal value about zero). Please note Wikipedia:Integrate_changes, which is part of the Wikipedia policy, requests that you integrate your changes so that they form a seamless part of the article. Loisel 02:14, 8 Jan 2004 (UTC)

text: I think that maybe this whole paragraph should be junked, since with the proper definition with 2 limits, there is no preferred point and no translation noninvariance. Thus this is not a deficiency of the Riemann integral.MarSch 15:30, 11 Mar 2005 (UTC)
thread_title:  Well done 

message: The definition of "almost everywhere" in https://en.wikipedia.org/wiki/Lebesgue_integration#Basic_theorems_of_the_Lebesgue_integral seems to be different
than the one in the article https://en.wikipedia.org/wiki/Almost_everywhere . The difference is that here the set where two functions differ has to have measure zero, but in the dedicated page it is enough that it is contained in a set of measure zero. In other words, the set where the functions differ does not have to be measurable in the second definition. Can someone confirm this? Thank you for a nice article otherwise. 129.194.183.160 (talk) 17:47, 24 September 2020 (UTC)
replies: text: Is the measure assumed to be the Lebesgue measure in this section? If not, then it is not necessarily true that subsets of null sets are measurable unless the measure space is complete. Tito Omburo (talk) 09:30, 22 May 2024 (UTC)
thread_title:  Almost everywhere