Talk:Lebesgue integral

3357:

chosen. In the Riemann integral we partition the x-axis and determine corresponding heights. In the Lebesgue integral we partition the y-axis and determine corresponding regions in the domain. So I believe a helpful diagram will somehow have to demonstrate a kind of “order of choices”. It would be especially useful when applied (1) to a continuous function, where there is obviously no difference between the Lebesgue integral and the Riemann integral, and (2) to the Dirichlet function, where you can more easily see the two different order of choices result in different integrals (especially how, by partitioning the codomain first, the preimage is not even a partition of the domain and therefore the quantities involved are clearly not “reachable” by the Riemann order of choices).

2730:

Perhaps you would like to clarify your understanding of the footnote and the definition in the cited source. I am guessing (without much evidence based on this interaction) that you have the competence required, although if you like we can enlist the help of someone at the math project. In the mean time have reverted your revert to restore the relevant, cited definition of the improper integral that applies to this case. Obviously another source of similar quality that defines the imroper integral so that the expression

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It's hair-splitting: the integral is clearly well-defined, and attempting to pin it down too much here runs the risk of original research (e.g., are Darboux sums kosher for unbounded integrands?) For proof you need look no further than the fact that you, a humble mathematician, were able to supply such a definition (by a non-standard and ad hoc approach). I have given a different approach that is sourced to secondary literature. I consider the matter settled.

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that corresponds to the heights of these rectangles. I know intuitions are, by definition, not perfectly rigorous or accurate. But this seems to be far enough from the reality of the Lebesgue integral, that it would actually cause more misunderstanding than understanding. Also, the diagram right below it seems to accurately represent the Lebesgue integral. So I vote we take down diagram with the misrepresentation.

3353:

x-axis. I see you mention that it is easy to correct the given picture, along these lines. That may be, although : I really do worry that a diagram which so fundamentally mis-matches the formalism will cause rather than resolve confusion. Perhaps if the diagram displays a prominent warning about this mis-match then it might be ok. But I’m not in control of the page, so I’ll let everyone else do what they will do.

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in the source space. This is not necessary and is nowehere used in the article.. Of course this means that if f = g a.e. and is measurable g may not be. There are a number of other changes which I don't see add anything to the article. For instance why add vector-valued functions here? If there is no objection I am going to revert. The only improvement was the use oof \liminf and \limsup whic I propose to keep.

840:]. The Lebesgue integral is defined by partitioning the range and adding up the contributions from the horizontal slabs of the partition. Apart from being horizontal, the main difference with the Riemann approach is that the slabs are of the form S× where S is a measurable set rather than an interval. I have put this section back in, since it addresses this very point. 33: 600:. 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. 664:

The diagram in the section titled "Intuitive Interpretation" seems misleading (these are the red rectangles underneath the blue ones). The approximation used by Lebesgue integration is not by length-wise rectangles. In particular, nothing in the approximations of the Lebesgue integral have a height

457:

The first sentence of the article talk about an area being bounded. If the fuction is discontinuous then no area is bounded. Of course I should also have made changes to say that the area is bounded from below by 0. I realize that even then you will have difficulty on the sides, so I will be bold and

3308:

If one wants to, it's probably fairly easy to see how an approximation as horizontal slabs can be converted into one expressed as simple functions, either geometrically by dropping some verticals onto the x-axis or perhaps algebraically by the summation by parts described above. This might be needed

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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

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which is in a rather obvious manner the sum of slabs obtained by partitioning the range. (This definition is proven equivalent in green Rudin, but I recently saw this as the primary definition given in Liptser and Shiryaev.) I think probably a source of confusion is that in the Lebesgue approach,

3348:

I write this having not lately looked at the changes or conversation in a while. But I would say that the problem isn’t the idea of partitioning the range, that is totally valid. Diagrams which display this are totally fine with me. This is indeed equivalent to the Lebesgue integral (I’ve proved

3317:

I had even more difficulty with the next diagram, containing the small rectangles of height dt. I'm may be too strongly affected by having originally learnt the construction of the Lebesgue integral via simple functions, but it might be more readable to take the shortest route towards this solution

2790:

Alternatively, we can agree to disagree. I think the expression is well-defined (without further elaboration!). (Although I am familiar with how to define improper integrals of functions singular at interior points, and many readers - and certain editors - appear not to be.) Since that seems to be

2309:

0. The upper and lower Darboux sums are both infinite in that case. So, by definition, so is the Riemann integral (at least according to the Darboux approach)." Sorry, yes, I did mean that notationwise. And no. The (proper) Riemann integral is defined for bounded functions on a finite interval, and

1558:

Please read the edit summary. I did not say the claims in this section are wrong, only that they are too heavy for an introduction. This section gives an alternative definition of Lebesgue integral, ... while preparing for the main one below? Interesting logic, isn't it? The point is: a supposedly

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Remarks on measurability condition.. MEasurability of a function as defined in the article is a property of the underlying sigma-algebras of source and target space. The user who made the latest edits, confused this with measurability of functions with respect to the sigma-algebra of the completion

2070:

0, the "proper" Riemann/Darboux integral is infinity, directly from the definition! There is no need to do any further analysis of that case. But since you seem to be uncomfortable with that, I included in a footnote a standard way of defining the improper integral in the presence of singularities

1892:

does not seem important here since it is not required to define the integral. In particular it is not present in the cited source. They define the integral as an improper Riemann integral without further elaboration, it being obvious that this can be defined without difficulty (to me at least). In

344:

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

3356:

I totally agree that I like diagrams which resemble the familiar Riemann picture but show the slight difference with the Lebesgue integral—-but I believe the important difference between them doesn’t easily show up in a static picture. The difference between them is the order in which points are

2275:

0. The upper and lower Darboux sums are both infinite in that case. So, by definition, so is the Riemann integral (at least according to the Darboux approach). But I think the more important point here is that the cited source didn't identify the issues with the definition that you seem to have.

340:

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

3352:

The issue is then diagramming the integral as horizontal rectangles with lateral sides determined by the partition, which I believe is not equivalent to the Lebesgue integral. To diagram it faithfully, the lateral sides of the rectangle should still go from some point on the curve, down to the

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Wow, apparently this started outright warfare. Sorry about that, I just thought that the depiction of length-wise rectangles deceptively makes it look like the width of the rectangles is due to the width of the cells of the partition on the vertical axis. But that's not true, the width of the

2729:

The definition in the cited source says that the (improper) integral of a positive function on a set A is obtained by cutting off the function above, integrating the bounded function, and then taking the limit as the cutoff tends to infinity. That is precisely what is happening in the footnote.

2554:

I'd agree that in general the Riemann integral doesn't like unbounded functions. It's just not that interesting in those cases, because the integral always fails to exist then. However, here the "proper" Riemann/Darboux integral diverges to infinity, which is not a problem. Anyway, that's not

628:

Any subset of a measure zero set has measure zero, so it doesn't make sense to talk about a subset of a measure zero set being non-measurable. A subset of a measure zero set may not be a Borel set, but it will always be Lebesgue measurable. I'm changing the language at the start of the "Basic

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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

3284:

Yes, I'd been wondering the same thing. I had difficulty with the length-wise rectangles in the diagram labelled "Lebesgue integration (in red)". Funnily enough, I think the edit war you're worried this may've started was probably quite constructive: the article seems better comparing

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As for rearranging the intuitive explanation part, I’m fairly neutral—-I only joined in editing that part because I thought the intuitive explanation that I encountered wasn’t intuitive enough. But if anyone wants to reorganize or change that for the better, I am all in favor of

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Regarding your example of a simple function, it can be verified as follows (although drawing a picture such as that in the relevant section of the article helps to make the required summation by parts a little clearer). To get the integral of a simple function

1574:

I've tried to extrapolate the informal part for the intuition section. The point is that the Lebesgue integral is really not that far removed from the intuitive explanation. I think the use of simple functions as the first approach seems to obscure that.

1591:

I'm not sure what the advantage is in bringing in the Lebesgue-Vitali theorem. Monotone functions are Riemann integrable, easily checked from the definition. Existence and interpretation of the improper Riemann integral in the section poses no difficulty.

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rectangles is determined by the distance all the way to the horizontal axis. I just worry that a picture suggesting a falsehood like this would mislead a person not yet familiar with the subject, in a way that's unnecessary.

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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

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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

2068:

I'm ok with that in a footnote, which seems like a reasonable compromise. But this insistence that one must do something special to interpret the improper integral is silly. If the distribution is infinite at some t:

1559:

informal intro cannot be as heavy or heavier than the main subject. Therefore I have moved the section past the main definition. As a side note, the article already has a section named "Intuitive interpretation".

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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

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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.

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So glad that we're in agreement. I've gone ahead and put back in the version that has a citation. If you would like to add your method of defining the integral, go ahead and restore it with a source. Thanks,

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makes sense would be welcome. But you seem to have reached the conclusion that it doesn't make sense, and therefore needs to be redefined in an ad hoc way to reach the bad cases. That requires a source!

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and the vertical sides of the square — running from (0,1) to (0,0) and from (1,1) to (1,0) — are not bounded by the graph. MarSch's change from "bounded" to "contained" solves the problem. --

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f(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∫

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to reconcile the diagram with the preceding quotation from Lebesgue that "I order the bills and coins according to identical values and then I pay the several heaps one after the other".

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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.

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It ties in with several key characteristics of Lebesgue's approach, eg. that the domain just needs to be a measure space (rather than requiring all the properties of the real line).

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the slabs are not rectangular. Anyway, the Lebesgue integral can be written as an equivalent (improper) Riemann integral, integrating up the horizontal (non-rectangular) slabs

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Intuitively, continuity is not necessary: Two disjoint rectangles with vertical sides and of different heights have an area and aren't representedby continuous functions.

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Please do not accuse editors acting in good faith (and discussing on the talk page!) of acting disruptively. That is not constructive. If you feel that I am disruptive,

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The other reason this source is unsuitable here (this time on terminological grounds) is the part of it that says "if this limit is finite, and in that case we say that

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oh, what is your "very good reason" for accusing another editor of disruption? I'll note that you're the one who changed the status quo here by section-blanking.

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Having re-read the article and the accompanying discussion, I can see some attractions in comparing the Lebesgue method to a contour map, ie. describing it as

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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.

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properties follow by continuity. Dieudonne disparages theRiemann integral (which in fact is the prevalent attitude among the mathematicians I know)

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You said "If you would like to add your method of defining the integral, go ahead and restore it with a source." There is no "my" method here. I

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Huh? Discontinuity implies unboundedness? What about the indicator function of the interval ? That's a bounded function with a discontinuity.

2664:

It might be a different page in your edition. Have you considered looking for "Improper integral" in either the index or the table of contents?

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I think we should try to keep the first sentence as easy as possible and I edited the article accordingly (bringing back the "2D bias"). --

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means with "unbounded" that the graph does not run along the boundary. For instance, with your indicator function, the graph looks like this

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and above the x-axis is about as unambiguous as you can get in natural language. And the anti 2D bias isn't very informative in my view.

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Stop disruptive editing. You are demonstrably wrong: there is a clear discrepancy between the source and your usage of it. The function

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Agreed. The red diagram is flat out wrong, not just misleading. It doesn't reflect the fact that, for simple non-negative functions,

1908:

Disruptive editing is frequently done is good faith, and I have a very good reason (see above) as to why you were being disruptive.

394:, which is part of the Knowledge policy, requests that you integrate your changes so that they form a seamless part of the article. 119: 80: 3153: 3581:{\displaystyle \int f=\lim _{n\to \infty }\sum _{k=0}^{n2^{n}}k2^{-n}\mu \{x|k2^{-n}<f(x)\leq (k+1)2^{-n}\}+n\mu \{x|f(x): --> 630: 254: 193: 2555:

important, since there is a standard definition of the improper integral in the literature that completely resolves this issue.

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For what it's worth, some authors actually do define the Lebesgue integral of a non-negative measurable function as the limit:

813:{\displaystyle \int _{S}f\,d\mu \,{\stackrel {\text{def}}{=}}\,\sum _{i}c_{i}\mu {\Bigl (}\{x\mid f(x)=c_{i}\}\cap S{\Bigr )}.} 1315: 315:

The Lebesgue approach is not the most elementary area-based integration theory; that distinction goes to the Riemann integral.

2052:" special and "≥" not. The answer, of course (and this is not immediately obvious), is that "≥" is just as special as ": --> 1893:

case readers find the idea of an improper Riemann integral too vexing, I have given a definition at length in a footnote.

573:. Your sentence would be perfect for integration. For those reading this article it is needlessly simplistic and ugly. - 341:

show that there are many common examples where the uniform convergence theorem is insufficient) should remain, I think.

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those functions only. This integral was not created with infinity in mind. That's what the improper one is for. So, if

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

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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).

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a relevant definition on both of these pages, but it's completely different and has nothing to do with what

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and the concept of (proper) Riemann integral is inapplicable. Therefore there is no improper one either.

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the improper Riemann integral does not always exist in this context, and that needs to be acknowledged;

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the standard definition. I can link to a relevant WP article if you like, and that will be my source.

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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the attitude of the original source as well, we can remove all explanations of its meaning here.

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0, the "proper" Riemann/Darboux integral is infinity, directly from the definition!". Well, if

2916:{\displaystyle f_{M}(x)={\begin{cases}f(x)&{\text{if}}\ x\leq M\\0&{\text{if}}\ x: --> 1781:, while insisting on something completely wrong (see above), constitutes disruptive behavior. 547: 505: 2573:

There is a well-defined limit according to at least one definition of the improper integral.

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The approach seems to be mathematically equivalent to the construction via simple functions.

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is not even Riemann-integrable. There is no improper Riemann integral in this case either.

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If it's "Elementary classical analysis" (1974), then p.269 does not have that definition.

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are distinct, positive, and arranged in (strictly) increasing order. By convention, set

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an alternative definition of improper Riemann integral because the ordinary one fits

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https://en.wikipedia.org/Lebesgue_integration#Basic_theorems_of_the_Lebesgue_integral

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which handles the special cases you seem to think requires some further analysis.

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you insist on is too restrictive and eliminates a huge number of important cases;

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The diagram in red is easy to understand and contrast with the Riemann approach.

2051:(a.e.) needs to be mentioned. Otherwise the reader will wonder what makes ": --> 574: 566: 485: 459: 426: 404: 395: 349: 124: 528: 450: 439: 101: 270: 206: 3655: 3370: 3339: 3271: 3256: 3237: 3217: 2800: 2720: 2689: 2673: 2658: 2644: 2629: 2614: 2597: 2582: 2564: 2543: 2285: 2267: 2080: 2062: 1932: 1917: 1902: 1790: 1601: 1584: 1568: 1552: 849: 832: 653: 638: 623: 609: 1777:

you reverted was both relevant and correct. Therefore reverting the edits

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change it to something which mentions area below a (positive) function.

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I would keep it, perhaps with a pointer to how it will be interpreted.

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p462 in the second edition, with Michael Hoffman (via Google books).

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And yes, the diagram underneath the red one makes a lot more sense.

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it in my spare time a while ago, so I’m fairly confident in this.).

388:(sometimes called the Improper Cauchy Principal value about zero) 2605:

What is the title, and what is the page number in this source?

1528:{\displaystyle \sum _{n=0}^{N}S_{n}=\sum _{n}(\mu \{x|f(x): --> 3208:

We need to allow the integral to be both finite and infinite.

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enables those ideas to be put into rigorous use. Great job.

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One can only overlook this difference if one chooses to.

2086:

You say, "If the distribution is infinite at some t: -->

2053:", and that's where the a.e.-equality becomes relevant. 3286: 2980: 2708: 2701: 2293: 1774: 3157: 1801:

is thataway. The added discussion of a.e. equality of

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the Riemann integral can be equivalently defined via

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c_{n+1}\})c_{n+1}=\sum _{n}\mu (A_{n})c_{n},}" /: -->

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c_{n+1}\})c_{n+1}=\sum _{n}\mu (A_{n})c_{n},}": -->

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The diagram with length-wise rectangles seems wrong

3638: 3580: 3200: 3142: 3113: 3076: 3053: 3026: 2969: 2939: 2915: 2779: 2526: 2476: 2411: 2372: 2248: 2183: 2151: 2041: 1884: 1838: 1760: 1700: 1638: 1527: 1296: 1162: 1098: 1012: 974: 941: 914: 812: 596:seems to be different than the one in the article 565:As far as I've seen there are also articles about 319:Don't think that's actually true (cf. Dieudonne's 915:{\displaystyle \sum _{n=1}^{N}c_{n}\chi _{A_{n}}} 802: 752: 3396: 3296:partitioning the range, rather than the domain. 3159: 2990: 3690:Knowledge level-5 vital articles in Mathematics 1770:This duality needs to be stated and explained; 2780:{\displaystyle \int _{0}^{\infty }f^{*}(t)dt} 2704:have. Thus, you don't have a relevant source. 2049:t\})=\mu (\{x\in E\mid f(x)\geq t\})}" /: --> 1530:c_{n+1}\})c_{n+1}=\sum _{n}\mu (A_{n})c_{n},} 386:I will modify the text to say something like 281:This page has archives. Sections older than 216:This page has archives. Sections older than 8: 3627: 3598: 3575: 3546: 3534: 3461: 2465: 2432: 2361: 2321: 2240: 2207: 2134: 2101: 2033: 2000: 1985: 1952: 1749: 1722: 1692: 1665: 1458: 1416: 1407: 1371: 1297:{\displaystyle \mu (A_{n})=\mu \{x|f(x): --> 1291: 1255: 1246: 1204: 1080: 1044: 791: 757: 614:Excellent point. This article is incorrect. 2696:I've looked at both. It's not there. There 1942:t\})=\mu (\{x\in E\mid f(x)\geq t\})}": --> 2249:{\displaystyle \mu (\{t\in E\mid f(t): --> 2152:{\displaystyle \mu (\{t\in E\mid f(t): --> 2042:{\displaystyle \mu (\{x\in E\mid f(x): --> 1701:{\displaystyle \mu (\{x\mid f(x)\geq t\})} 1163:{\displaystyle \Delta y_{n}=c_{n+1}-c_{n}} 598:https://en.wikipedia.org/Almost_everywhere 69: 3604: 3596: 3552: 3525: 3479: 3467: 3449: 3434: 3426: 3415: 3399: 3384: 3188: 3178: 3162: 3155: 3135: 3096: 3090: 3066: 3045: 3039: 3006: 2988: 2953: 2932: 2890: 2865: 2843: 2825: 2819: 2756: 2746: 2741: 2735: 2512: 2491: 2477:{\displaystyle \mu \{x\in E\mid f(x): --> 2427: 2394: 2388: 2373:{\displaystyle \mu \{x\in E\mid f(x): --> 2355: 2316: 2199: 2167: 2093: 1944: 1853: 1807: 1714: 1657: 1616: 1516: 1503: 1487: 1468: 1446: 1422: 1401: 1377: 1359: 1346: 1336: 1325: 1319: 1285: 1261: 1234: 1210: 1189: 1177: 1154: 1135: 1122: 1113: 1090: 1074: 1050: 1032: 1026: 987: 960: 954: 933: 927: 904: 899: 889: 879: 868: 862: 801: 800: 785: 751: 750: 741: 731: 717: 712: 710: 709: 690: 684: 592:The definition of "almost everywhere" in 2635:Only p.269 is cited. There is no title. 3680:Knowledge vital articles in Mathematics 3322:and its accompanying diagram into the 725: 707: 699: 331:Rare? A couple of derivatives will do. 71: 30: 1611:as I previously stated, the condition 1099:{\displaystyle S_{n}=\mu \{x|f(x): --> 327:Uniform convergence of Fourier series. 291:when more than 5 sections are present. 226:when more than 5 sections are present. 3695:B-Class vital articles in Mathematics 3318:and move the final paragraphs of the 2307:you say "I think you mean μ(f(x): --> 2043:t\})=\mu (\{x\in E\mid f(x)\geq t\})} 1761:{\displaystyle \mu (\{x\mid f(x): --> 7: 117:This article is within the scope of 3228:Ah you're right about that source. 629:theorems" section to reflect this. 369:I just removed the following text: 60:It is of interest to the following 3705:High-priority mathematics articles 3406: 3169: 2747: 2471: 2367: 2143: 1639:{\displaystyle \mu (E)<\infty } 1633: 1310:so that summation by parts gives: 1115: 1083: 496:------ -------- ------ 25: 3639:{\displaystyle \{t|f(t)<x\}dx} 285:may be automatically archived by 220:may be automatically archived by 137:Knowledge:WikiProject Mathematics 3675:Knowledge level-5 vital articles 3150:is integrable", in reference to 3027:{\displaystyle \min(A,f^{*}(t))} 1885:{\displaystyle \mu (f(x)\geq t)} 1607:Please stop disruptive editing: 236: 183: 140:Template:WikiProject Mathematics 104: 94: 73: 40: 31: 157:This article has been rated as 3685:B-Class level-5 vital articles 3618: 3612: 3605: 3566: 3560: 3553: 3518: 3506: 3500: 3494: 3468: 3403: 3257:13:53, 27 September 2021 (UTC) 3238:13:17, 27 September 2021 (UTC) 3218:12:36, 27 September 2021 (UTC) 3166: 3021: 3018: 3012: 2993: 2860: 2854: 2837: 2831: 2801:11:17, 27 September 2021 (UTC) 2768: 2762: 2721:00:33, 27 September 2021 (UTC) 2690:22:05, 26 September 2021 (UTC) 2674:21:09, 26 September 2021 (UTC) 2659:21:05, 26 September 2021 (UTC) 2645:21:01, 26 September 2021 (UTC) 2630:20:58, 26 September 2021 (UTC) 2615:20:51, 26 September 2021 (UTC) 2598:20:44, 26 September 2021 (UTC) 2583:20:37, 26 September 2021 (UTC) 2565:21:04, 26 September 2021 (UTC) 2544:20:07, 26 September 2021 (UTC) 2518: 2499: 2456: 2450: 2345: 2339: 2286:15:57, 26 September 2021 (UTC) 2268:15:38, 26 September 2021 (UTC) 2243: 2231: 2225: 2204: 2137: 2125: 2119: 2098: 2081:15:20, 26 September 2021 (UTC) 2063:14:54, 26 September 2021 (UTC) 2036: 2024: 2018: 1997: 1988: 1976: 1970: 1949: 1933:15:05, 26 September 2021 (UTC) 1918:14:29, 26 September 2021 (UTC) 1903:11:58, 26 September 2021 (UTC) 1879: 1870: 1864: 1858: 1833: 1824: 1818: 1812: 1791:02:17, 26 September 2021 (UTC) 1752: 1740: 1734: 1719: 1695: 1683: 1677: 1662: 1627: 1621: 1602:00:25, 26 September 2021 (UTC) 1585:14:14, 24 September 2021 (UTC) 1569:13:32, 24 September 2021 (UTC) 1553:12:30, 23 September 2021 (UTC) 1509: 1496: 1461: 1436: 1430: 1423: 1391: 1385: 1378: 1365: 1275: 1269: 1262: 1224: 1218: 1211: 1195: 1182: 1064: 1058: 1051: 850:11:41, 23 September 2021 (UTC) 775: 769: 624:00:54, 25 September 2020 (UTC) 610:17:47, 24 September 2020 (UTC) 1: 3328:Via improper Riemann integral 1839:{\displaystyle \mu (f(x): --> 417:Wed Jan 14 03:09:34 UTC 2004 131:and see a list of open tasks. 3700:B-Class mathematics articles 3656:11:00, 30 January 2024 (UTC) 3371:19:25, 19 January 2022 (UTC) 3340:14:07, 19 January 2022 (UTC) 3272:02:45, 30 October 2021 (UTC) 2273:I think you mean μ(f(x): --> 1106:c_{n}\}\Delta y_{n}}" /: --> 1013:{\displaystyle n=0,\dots ,N} 445:Integral as area under curve 2620:It's cited in the article. 1305:c_{n-1}\}-\mu \{x|f(x): --> 1298:c_{n-1}\}-\mu \{x|f(x): --> 1174:c_{n-1}\}-\mu \{x|f(x): --> 833:20:22, 13 August 2021 (UTC) 392:Knowledge:Integrate_changes 3721: 1170:. Note that we may write 1024:c_{n}\}\Delta y_{n}}": --> 3114:{\displaystyle f^{*}: --> 2412:{\displaystyle t_{0}: --> 1536:c_{n}\}-\mu \{x|f(x): --> 1529:c_{n}\}-\mu \{x|f(x): --> 1316:c_{n}\}-\mu \{x|f(x): --> 453:15:29, 11 Mar 2005 (UTC) 337:07:24, 18 Dec 2003 (UTC) 323:ona sub-Riemann theory). 214: 181: 156: 89: 68: 18:Talk:Lebesgue integration 3320:Intuitive interpretation 2380:t_{0}\}=\infty }" /: --> 838:Please read the section 654:09:30, 22 May 2024 (UTC) 639:21:31, 21 May 2024 (UTC) 550:15:10, 14 Mar 2005 (UTC) 531:14:26, 14 Mar 2005 (UTC) 523:Referring to the region 508:14:12, 14 Mar 2005 (UTC) 473:00:09, 14 Mar 2005 (UTC) 462:15:40, 11 Mar 2005 (UTC) 429:08:05, 1 Feb 2004 (UTC) 407:15:30, 11 Mar 2005 (UTC) 398:02:14, 8 Jan 2004 (UTC) 352:02:04, 8 Jan 2004 (UTC) 163:project's priority scale 3289:a bit before and after. 3230:2600:387:F:4917:0:0:0:6 2793:2600:387:F:4917:0:0:0:6 1773:the information in the 975:{\displaystyle c_{0}=0} 577:14:40, 4 Apr 2005 (UTC) 120:WikiProject Mathematics 3670:B-Class vital articles 3640: 3583: 3441: 3202: 3144: 3116: 3078: 3055: 3028: 2972: 2941: 2923:M\end{cases}}}" /: --> 2918: 2781: 2528: 2527:{\displaystyle t\in ,} 2479: 2414: 2375: 2314:t_{0}\}=\infty }": --> 2251: 2186: 2159:0\})=\infty ,}" /: --> 2154: 2044: 1886: 1841: 1763: 1702: 1640: 1531: 1341: 1300: 1164: 1101: 1014: 982:. Then consider, for 976: 943: 916: 884: 814: 288:Lowercase sigmabot III 223:Lowercase sigmabot III 3641: 3584: 3411: 3203: 3145: 3117: 3079: 3056: 3054:{\displaystyle f^{*}} 3029: 2973: 2970:{\displaystyle x: --> 2942: 2919: 2782: 2529: 2480: 2415: 2376: 2252: 2187: 2184:{\displaystyle t: --> 2155: 2045: 1887: 1842: 1764: 1703: 1641: 1532: 1321: 1301: 1165: 1102: 1015: 977: 944: 942:{\displaystyle c_{n}} 917: 864: 815: 348:Anyway, have a ball. 47:level-5 vital article 3595: 3383: 3154: 3134: 3089: 3065: 3038: 2987: 2952: 2931: 2818: 2817:M\end{cases}}}": --> 2734: 2490: 2484:t\}=\infty }" /: --> 2426: 2387: 2315: 2308:t)=∞ for some t: --> 2274:t)=∞ for some t: --> 2198: 2166: 2092: 2091:0\})=\infty ,}": --> 1943: 1852: 1806: 1713: 1656: 1615: 1318: 1176: 1112: 1100:c_{n}\}\Delta y_{n}} 1025: 986: 953: 926: 861: 683: 321:Treatise on Analysis 143:mathematics articles 2927:in the source cuts 2751: 571:Riemann integration 3636: 3578: 3410: 3198: 3197: 3173: 3140: 3111: 3077:{\displaystyle A,} 3074: 3051: 3024: 2967: 2947:down to zero when 2937: 2913: 2908: 2777: 2737: 2524: 2474: 2425:t\}=\infty }": --> 2409: 2370: 2246: 2181: 2149: 2039: 1882: 1836: 1758: 1698: 1636: 1525: 1492: 1364: 1294: 1160: 1096: 1010: 972: 939: 912: 810: 736: 726: 708: 700: 112:Mathematics portal 56:content assessment 3395: 3158: 3143:{\displaystyle f} 2940:{\displaystyle f} 2896: 2893: 2871: 2868: 1483: 1355: 727: 722: 720: 588:Almost everywhere 295: 294: 260: 259: 230: 229: 177: 176: 173: 172: 169: 168: 16:(Redirected from 3712: 3645: 3643: 3642: 3637: 3608: 3589: 3586: 3585: 3579: 3556: 3533: 3532: 3487: 3486: 3471: 3457: 3456: 3440: 3439: 3438: 3425: 3409: 3283: 3249:StrokeOfMidnight 3210:StrokeOfMidnight 3207: 3205: 3204: 3199: 3193: 3192: 3183: 3182: 3172: 3149: 3147: 3146: 3141: 3122: 3119: 3118: 3112: 3101: 3100: 3083: 3081: 3080: 3075: 3060: 3058: 3057: 3052: 3050: 3049: 3033: 3031: 3030: 3025: 3011: 3010: 2978: 2975: 2974: 2968: 2946: 2944: 2943: 2938: 2924: 2921: 2920: 2914: 2912: 2911: 2895: 2894: 2891: 2870: 2869: 2866: 2830: 2829: 2786: 2784: 2783: 2778: 2761: 2760: 2750: 2745: 2713:StrokeOfMidnight 2651:StrokeOfMidnight 2637:StrokeOfMidnight 2607:StrokeOfMidnight 2536:StrokeOfMidnight 2533: 2531: 2530: 2525: 2517: 2516: 2485: 2482: 2481: 2475: 2420: 2417: 2416: 2410: 2399: 2398: 2381: 2378: 2377: 2374:t_{0}\}=\infty } 2371: 2360: 2359: 2260:StrokeOfMidnight 2257: 2254: 2253: 2247: 2192: 2189: 2188: 2182: 2160: 2157: 2156: 2150: 2055:StrokeOfMidnight 2050: 2047: 2046: 2040: 1910:StrokeOfMidnight 1891: 1889: 1888: 1883: 1847: 1844: 1843: 1837: 1783:StrokeOfMidnight 1769: 1766: 1765: 1759: 1707: 1705: 1704: 1699: 1645: 1643: 1642: 1637: 1561:StrokeOfMidnight 1538: 1534: 1533: 1526: 1521: 1520: 1508: 1507: 1491: 1479: 1478: 1457: 1456: 1426: 1406: 1405: 1381: 1363: 1351: 1350: 1340: 1335: 1307: 1306:c_{n}\}}" /: --> 1303: 1302: 1295: 1290: 1289: 1265: 1245: 1244: 1214: 1194: 1193: 1169: 1167: 1166: 1161: 1159: 1158: 1146: 1145: 1127: 1126: 1107: 1104: 1103: 1097: 1095: 1094: 1079: 1078: 1054: 1037: 1036: 1019: 1017: 1016: 1011: 981: 979: 978: 973: 965: 964: 948: 946: 945: 940: 938: 937: 921: 919: 918: 913: 911: 910: 909: 908: 894: 893: 883: 878: 825:StrokeOfMidnight 819: 817: 816: 811: 806: 805: 790: 789: 756: 755: 746: 745: 735: 724: 723: 721: 718: 716: 711: 695: 694: 616:StrokeOfMidnight 359:Charles Matthews 335:Charles Matthews 290: 274: 251: 250: 240: 232: 225: 209: 187: 179: 145: 144: 141: 138: 135: 114: 109: 108: 98: 91: 90: 85: 77: 70: 53: 44: 43: 36: 35: 27: 21: 3720: 3719: 3715: 3714: 3713: 3711: 3710: 3709: 3660: 3659: 3593: 3592: 3521: 3475: 3445: 3430: 3380: 3379: 3277: 3184: 3174: 3152: 3151: 3132: 3131: 3092: 3086: 3085: 3063: 3062: 3041: 3036: 3035: 3034:merely freezes 3002: 2985: 2984: 2949: 2948: 2929: 2928: 2907: 2906: 2888: 2882: 2881: 2863: 2844: 2821: 2815: 2814: 2752: 2732: 2731: 2508: 2488: 2487: 2423: 2422: 2390: 2384: 2383: 2351: 2312: 2311: 2195: 2194: 2163: 2162: 2089: 2088: 1940: 1939: 1850: 1849: 1803: 1802: 1710: 1709: 1654: 1653: 1613: 1612: 1512: 1499: 1464: 1442: 1397: 1342: 1314: 1313: 1281: 1230: 1185: 1172: 1171: 1150: 1131: 1118: 1110: 1109: 1086: 1070: 1028: 1022: 1021: 984: 983: 956: 951: 950: 929: 924: 923: 900: 895: 885: 859: 858: 781: 737: 686: 681: 680: 662: 602:129.194.183.160 590: 525:under the curve 497: 447: 435: 381: 377: 302: 286: 275: 269: 245: 221: 210: 205: 142: 139: 136: 133: 132: 110: 103: 83: 54:on Knowledge's 51: 41: 23: 22: 15: 12: 11: 5: 3718: 3716: 3708: 3707: 3702: 3697: 3692: 3687: 3682: 3677: 3672: 3662: 3661: 3635: 3632: 3629: 3626: 3623: 3620: 3617: 3614: 3611: 3607: 3603: 3600: 3577: 3574: 3571: 3568: 3565: 3562: 3559: 3555: 3551: 3548: 3545: 3542: 3539: 3536: 3531: 3528: 3524: 3520: 3517: 3514: 3511: 3508: 3505: 3502: 3499: 3496: 3493: 3490: 3485: 3482: 3478: 3474: 3470: 3466: 3463: 3460: 3455: 3452: 3448: 3444: 3437: 3433: 3429: 3424: 3421: 3418: 3414: 3408: 3405: 3402: 3398: 3394: 3391: 3388: 3376: 3375: 3374: 3373: 3358: 3354: 3350: 3343: 3342: 3315: 3312: 3311: 3310: 3306: 3303: 3300: 3290: 3245: 3244: 3243: 3242: 3241: 3240: 3221: 3220: 3196: 3191: 3187: 3181: 3177: 3171: 3168: 3165: 3161: 3139: 3128: 3124: 3123: 3110: 3107: 3104: 3099: 3095: 3073: 3070: 3048: 3044: 3023: 3020: 3017: 3014: 3009: 3005: 3001: 2998: 2995: 2992: 2966: 2963: 2960: 2957: 2936: 2925: 2917:M\end{cases}}} 2910: 2905: 2902: 2899: 2889: 2887: 2884: 2883: 2880: 2877: 2874: 2864: 2862: 2859: 2856: 2853: 2850: 2849: 2847: 2842: 2839: 2836: 2833: 2828: 2824: 2811: 2810: 2806: 2805: 2804: 2803: 2788: 2776: 2773: 2770: 2767: 2764: 2759: 2755: 2749: 2744: 2740: 2724: 2723: 2705: 2693: 2692: 2677: 2676: 2633: 2632: 2603: 2602: 2601: 2600: 2571: 2570: 2569: 2568: 2567: 2547: 2546: 2523: 2520: 2515: 2511: 2507: 2504: 2501: 2498: 2495: 2473: 2470: 2467: 2464: 2461: 2458: 2455: 2452: 2449: 2446: 2443: 2440: 2437: 2434: 2431: 2408: 2405: 2402: 2397: 2393: 2369: 2366: 2363: 2358: 2354: 2350: 2347: 2344: 2341: 2338: 2335: 2332: 2329: 2326: 2323: 2320: 2305: 2296:it, you don't 2289: 2288: 2245: 2242: 2239: 2236: 2233: 2230: 2227: 2224: 2221: 2218: 2215: 2212: 2209: 2206: 2203: 2180: 2177: 2174: 2171: 2153:0\})=\infty ,} 2148: 2145: 2142: 2139: 2136: 2133: 2130: 2127: 2124: 2121: 2118: 2115: 2112: 2109: 2106: 2103: 2100: 2097: 2084: 2083: 2038: 2035: 2032: 2029: 2026: 2023: 2020: 2017: 2014: 2011: 2008: 2005: 2002: 1999: 1996: 1993: 1990: 1987: 1984: 1981: 1978: 1975: 1972: 1969: 1966: 1963: 1960: 1957: 1954: 1951: 1948: 1938:The fact that 1936: 1935: 1906: 1905: 1881: 1878: 1875: 1872: 1869: 1866: 1863: 1860: 1857: 1835: 1832: 1829: 1826: 1823: 1820: 1817: 1814: 1811: 1794: 1793: 1771: 1768:t\}).}" /: --> 1757: 1754: 1751: 1748: 1745: 1742: 1739: 1736: 1733: 1730: 1727: 1724: 1721: 1718: 1697: 1694: 1691: 1688: 1685: 1682: 1679: 1676: 1673: 1670: 1667: 1664: 1661: 1650: 1647: 1635: 1632: 1629: 1626: 1623: 1620: 1605: 1604: 1588: 1587: 1577:164.52.242.130 1556: 1555: 1545:164.52.242.130 1541: 1540: 1539: 1524: 1519: 1515: 1511: 1506: 1502: 1498: 1495: 1490: 1486: 1482: 1477: 1474: 1471: 1467: 1463: 1460: 1455: 1452: 1449: 1445: 1441: 1438: 1435: 1432: 1429: 1425: 1421: 1418: 1415: 1412: 1409: 1404: 1400: 1396: 1393: 1390: 1387: 1384: 1380: 1376: 1373: 1370: 1367: 1362: 1358: 1354: 1349: 1345: 1339: 1334: 1331: 1328: 1324: 1308: 1293: 1288: 1284: 1280: 1277: 1274: 1271: 1268: 1264: 1260: 1257: 1254: 1251: 1248: 1243: 1240: 1237: 1233: 1229: 1226: 1223: 1220: 1217: 1213: 1209: 1206: 1203: 1200: 1197: 1192: 1188: 1184: 1181: 1175:c_{n}\}}": --> 1157: 1153: 1149: 1144: 1141: 1138: 1134: 1130: 1125: 1121: 1117: 1093: 1089: 1085: 1082: 1077: 1073: 1069: 1066: 1063: 1060: 1057: 1053: 1049: 1046: 1043: 1040: 1035: 1031: 1020:, the "slab" 1009: 1006: 1003: 1000: 997: 994: 991: 971: 968: 963: 959: 936: 932: 907: 903: 898: 892: 888: 882: 877: 874: 871: 867: 853: 852: 842:164.52.242.130 821: 820: 809: 804: 799: 796: 793: 788: 784: 780: 777: 774: 771: 768: 765: 762: 759: 754: 749: 744: 740: 734: 730: 715: 706: 703: 698: 693: 689: 661: 658: 657: 656: 589: 586: 585: 584: 583: 582: 581: 580: 579: 578: 556: 555: 554: 553: 552: 551: 539: 538: 537: 536: 535: 534: 533: 532: 514: 513: 512: 511: 510: 509: 495: 494: 493: 492: 491: 490: 489: 477: 476: 475: 474: 464: 463: 446: 443: 434: 431: 410: 409: 408: 390:. Please note 379: 375: 367: 366: 365: 364: 363: 329: 328: 317: 316: 301: 298: 293: 292: 280: 277: 276: 271: 267: 265: 262: 261: 258: 257: 247: 246: 241: 235: 228: 227: 215: 212: 211: 203: 201: 198: 197: 189: 175: 174: 171: 170: 167: 166: 155: 149: 148: 146: 129:the discussion 116: 115: 99: 87: 86: 78: 66: 65: 59: 37: 24: 14: 13: 10: 9: 6: 4: 3: 2: 3717: 3706: 3703: 3701: 3698: 3696: 3693: 3691: 3688: 3686: 3683: 3681: 3678: 3676: 3673: 3671: 3668: 3667: 3665: 3658: 3657: 3653: 3649: 3633: 3630: 3624: 3621: 3615: 3609: 3601: 3572: 3569: 3563: 3557: 3549: 3543: 3540: 3537: 3529: 3526: 3522: 3515: 3512: 3509: 3503: 3497: 3491: 3488: 3483: 3480: 3476: 3472: 3464: 3458: 3453: 3450: 3446: 3442: 3435: 3431: 3427: 3422: 3419: 3416: 3412: 3400: 3392: 3389: 3386: 3372: 3368: 3364: 3359: 3355: 3351: 3347: 3346: 3345: 3344: 3341: 3337: 3333: 3329: 3325: 3321: 3316: 3313: 3307: 3304: 3301: 3298: 3297: 3295: 3291: 3288: 3281: 3276: 3275: 3274: 3273: 3269: 3265: 3259: 3258: 3254: 3250: 3239: 3235: 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