Talk:Lebesgue integral

Hello! I think this article is looking very good. Over the past several months many improvements have been made. I wonder if it would be better titled "Lebesgue integral", since (if I'm not mistaken) that term is rather more common in text books. Certainly "Lebesgue integral" is more common than "Lebesgue integration" on the web as shown by Google searches. Likewise "Lebesgue-Stieltjes integral" is more common on the web than "Lebesgue-Stieltjes integration". There do exist Lebesgue integral and Lebesgue-Stieltjes integral in WP but these are redirects. However, Riemann integral is an article and Riemann integration is a redirect; also integral is an article and integration is a disambiguation page (there is no integration (mathematics) article). So: in summary, I propose that we move Lebesgue integration to Lebesgue integral and likewise with Lebesgue-Stieltjes. I look forward to your comments. Regards & happy editing, Wile E. Heresiarch 16:57, 21 Jun 2004 (UTC)

On the whole, page names migrate towards the more abstract term. I think this is more suitable for WP, really: so that e.g. nerve is thought of under nervous system first (haven't checked the actual status of those). So I'd be happy to leave it as Lebesgue integration. Charles Matthews 18:08, 21 Jun 2004 (UTC)

Hmm, I guess I'm not convinced. The Lebesgue integral is the object of interest, and the purpose of Lebesgue integration is to construct the Lebesgue integral, is it not? What other items are considered under the general heading of Lebesgue integration? Perhaps I don't get out enough, of course; I wouldn't be at all surprised. Happy editing, Wile E. Heresiarch 23:31, 22 Jun 2004 (UTC)

Maybe one way would be to have an article like integration theories in mathematical analysis, as a sort of umbrella; and then call the articles on particular theories Riemann integral, Lebesgue integral and so on. Well, such an article would be useful in itself, as a survey. Charles Matthews 17:05, 23 Jun 2004 (UTC)

Hello. About notation, under "Equivalent formulations" there are L¹ and C_c. Do these want to be italicized as L¹ and C_c or maybe C_c ? I'm just wondering how these are conventionally presented. Happy editing, Wile E. Heresiarch 14:14, 13 Apr 2004 (UTC)

I think Lang and Rudin both use L¹ and C_c. Loisel 06:21, 16 Apr 2004 (UTC)

Hello, I've reworked the "Introduction" section to situate the Lebesgue integral historically, and to summarize the important differences with the Riemann integral. Then there are definitions and properties, and then there are the more detailed discussion sections. I hope this organization address the concerns with the "Rudinesque" previous revision. FWIW I'm not really happy with the "Lebesgue vs Riemann" aspect of this article, which might suggest there are exactly 2 defns of the integral; it's more "Lebesgue vs every predecessor". Also, I think "failure of the Riemann integral" gives a mistaken impression -- the Riemann integral has been a big success overall. Well, hope this helps. Wile E. Heresiarch 20:09, 8 Feb 2004 (UTC)

All I know is, after I studied the Riemann integral as an undergrad, I thought it was pretty hot stuff, so when I heard about this thing called Lebesgue integration, it took quite a bit of argument to convince me that this new-fangled thing was even necessary in the first place. The introduction is a good start, but I would take select parts of the "discussion" and merge them. As for giving the impression there are 2 defs of "integral", this is more a misunderstanding of math in general...people think there is some Platonic concept called "the integral" that mathematicians go out and "discover" its "true nature"...when there are just various definitions and theories that just are what they are. It's very difficult to explain this. Maybe it should be pointed out that the Riemann and Lebesgue integrals aren't the only theories of integration that have been developed -- just the most popular and widely used. Revolver 22:56, 8 Feb 2004 (UTC)

I don't think the discussion should come after the formal definitions and results. If someone is unfamiliar with the Lebesgue integral (and most will be who read the article) then the formal definition will seem pointless without some kind of motivation or discussion. As the article is now, someone who doesn't know the Lebesgue integral is likely to just give up part-way through the formal derivations and never reach the discussion. We're not trying to imitate Rudin. Revolver 02:06, 7 Feb 2004 (UTC)

If the definitions are incomprehensible, the discussion is irrelevant (and likewise incomprehensible). The most you can say to someone who doesn't understand the definition is "All definitions of the integral are the same for easy cases, but the Lebesgue integral handles some more difficult cases."

Well, before the Lebesgue integral was developed, many mathematicians discussed why they thought a new type of integral was necessary, what problems it should attack, what specific examples motivated it. The idea that it's impossible to "discuss" a mathematical subject without going through the details of the definitions doesn't seem right to me -- again, if for no other reason than that the mathematicians did it themselves for years before arriving the modern definition (think of groups, topological spaces, non-Euclidean geometries, etc.) I think it's quite possible to explain the specific issues that eventually led to the creation of the Lebesgue integral, without defining the Lebesgue integral itself. And yes, I'll try to do this and put it on the talk page someday...after I finish my thesis, find a job, move, etc.,...;-) Revolver 22:56, 8 Feb 2004 (UTC)

I think the (short) section just before the definition should say that. A secondary problem is that the discussion, as it stands, is needlessly verbose; it needs a lot more focus. Wile E. Heresiarch 09:20, 7 Feb 2004 (UTC)

What should ideally be done, is that the "discussion" and the "formal definition" should become merged into a single thread of narration, not separated into two pieces. This of course, would take a good deal of thought and effort to do it right. Revolver 02:08, 7 Feb 2004 (UTC)

I'm not convinced that is desirable. Maybe you can put a version of this kind of presentation on your talk page and have people look at it. Just a thought. Wile E. Heresiarch 09:20, 7 Feb 2004 (UTC)

Wile, see below, there's a guy who claims this article is too advanced. The text now under "Discussion" was my attempt at answering that gripe.

Loisel 21:13, 6 Feb 2004 (UTC)

Hello Loisel, thanks for bringing up this topic. I moved the discussion sections below the definitions, etc., because the discussion is quite long winded and goes off on several tangents. That's fine, really, since the Lebesgue integral is important and there's a lot to be said about it. But the article works better as a reference if we get right to the point.

Maybe to address the concern that the article is too abstract, we can make the "Introduction" section (just before the defn) better (not necessarily longer). Without getting technical, one can say the Lebesgue integral is more general than some other defns, and also the it's defined as the limit of a sequence of simpler integrals. Perhaps an example of a sequence of simple functions will make a good companion article. Wile E. Heresiarch 01:49, 7 Feb 2004 (UTC)

Michael, why did you remove the previous article and leave a corpse in its stead? I've now restored the article.

Loisel 08:04, 3 Feb 2004 (UTC)

I've made a correction to the Technical Difficulties regarding improper Reimann Integrals on this page. What was defined was not the Improper Reimann Integral but the improper Cauchy Pricipal Value. The improper Reimann Integral does not exist in this example.

-joshua

Look - I'm sure that this is a well-writen article, but could someone (knowledgeable) put in a 2-3 line explanation of what it actually about, and what it is used for? The opening sentence "Let m be ..." isn't really au fait in a generalist encyclopedia. I'm really glad we have this sort of advanced stuff, but in this case I don't even understand what I don't understand. - MMGB

Shouldn't it be at some point made clear that m is the Lebesgue measure? Otherwise, there's no guarantee the integral will correspond to Riemann integration - if m is the counting measure, it will correspond to summation. But I'm not sure where precisely to work that in.

The link to monotone convergence theorem is to a different theorem.

Charles Matthews 14:00, 6 Sep 2003 (UTC)

Fixed. Loisel 06:38, 7 Sep 2003 (UTC)

Too advanced!

This encyclopedia article is too advanced; the only people who will be able to get the meaning are those who have already learned Lebesgue theory thoroughly. The modern, extremely compact, definitions of mathematics are simply a shorthand for people already familiar with the underlying concepts and some illustrative examples; without including the latter, the former is unintelligible.

In fairness, an article on the Lebesgue integral is certainly only of interest to a specialized audience. And it would perhaps be appropriate to include the existing article as an advanced appendix as a minimal, extremely formal, refresher, less formal explanation is needed for anyone who doesn't remember or hasn't learned what ideas the formal definition corresponds to.

For the main body of the article, it would be nice to follow history somewhat: (1) explanation of the problems encountered with the Riemann integral including specific functions or applications (2) elementary definition (NOT a formal argument: emphasis on underlying IDEAS, not on formal definitions) and explanation of how the problems of Riemann are resolved by Lebesgue (that is, examples) (3) perhaps a little modern history on the growth of analysis since Lebesgue. The formalities could be a conclusion of sorts. I would write such an article myself, but I don't know Lebesgue theory sufficiently well (hence my need to refer to an encyclopedia article); perhaps someone else might? (Anon.)

A page on improper integrals in the Riemann theory would be a help. But no one-page intro to the Lebesgue integral is going to be easy.

Charles Matthews 16:10, 23 Oct 2003 (UTC)

This is a valid point, however I'm not very good at history. I could attempt to vulgarize the concepts discussed in the "formal construction" section, but I couldn't color the text with historical anecdotes about integration theory: I don't know any.

Would that help?

Loisel 16:47, 24 Oct 2003 (UTC)

Is this better? Loisel 18:15, 24 Oct 2003 (UTC)

I'm sorry, this topic is advanced pdenapo Wed Jan 14 03:09:34 UTC 2004

Well done

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.

--

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)

About the uniform convergence: it is true that twice differentiable periodic functions have uniformly convergent Fourier series. This is a very thin set in L^2. That is one of the meanings of "rare." Feel free to adjust the wording if you think you can improve it, but the goal of that passage (to show that there are many common examples where the uniform convergence theorem is insufficient) should remain, I think.

About "elementary." I meant to recognize the order in which this is usually taught. I don't know the sub-Riemann theory you mention, but I should've thought there would be a gazillion variants that claim to be more elementary than the Riemann integral. Again, feel free to adjust the wording if you think you can improve it. If you're changing it, I think it should probably say that Riemann is "more elementary" than Lebesgue, but I doubt it should refer to some obscure theory that's not generally taught.

Anyway, have a ball. Loisel 02:04, 8 Jan 2004 (UTC)

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∫_a^bf(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)

I've rewriten this article since the old version focused only on the technical difficulties of Riemman integral, rather than defining the concept of Lebesgue integral. I've included the examples in the old version, though pdenapo Wed Jan 14 03:09:34 UTC 2004

The new introduction is nice, but not consitent with the definition that appears below. [User:pdenapo|pdenapo]]Wed Jan 14 03:09:34 UTC 2004

I think the rewrite has destroyed a large quantity of useful information and has generally decreased the quality of the article. I'm tempted to revert.

Loisel 08:05, 1 Feb 2004 (UTC)

Equivalent Formulations

I think this page is very good. Hello, Do you have references for the equivalent formulations of the integral? (unique continuous extension of a linear functional...) Thanks Sergei Vieira [email protected]