When students take a course on real analysis, they have likely learned about Riemann integrals.

What is a good motivation why they have to learn a new way to integrate?

A student don't want to hear to something like "You will now learn the new concept and in next years you will the importance of it". It should be some understandable example for the student (It is also okay to give an example after some weeks of the real analysis class, but avoid answers which a student at that level will not understand after the whole lecture.)

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    In relation to this question I highly recommend David Bressoud's book A Radical Approach to Lebesgue's Theory of Integration, which discusses the historical motivations for the theory. Another nice reference is the chapter on Lebesgue in William Dunham's The Calculus Gallery, which again frames the birth of Lebesgue integration in terms of the broader development of real analysis through the 19th century. – benblumsmith Mar 14 '14 at 19:47
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    I second the recommendation for A Radical Approach to Lebesgue's Theory of Integration. I used this book a lot for motivation and historical background for preparing my lectures on the Lebesgue integral in second-semester analysis. – Jim Belk Mar 14 '14 at 23:05
  • There is another extreme, which says the Riemann integral should not be taught at all, but rather we should use the Lebesgue integral from the start. – Gerald Edgar Oct 7 at 13:01
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What is striking about the Lebesgue integral is how relatively nicely limits play together with this integral, things like the Dominated convergenece theorem are great. I think one can appreciate this result (especially when contrasted with the more clumsy Riemann Integral analogues) right away. One could mention this right at the start, before everything else.

Put differently, that it is desirable to have $\int \lim f_n = \lim \int f_n $ be true (and make sense) seems essentially self-evident. The 'new' integral is there to make this true under weaker/nicer assumptions (on $f_n$).

By contrast to me the 'one can integrate more functions' reason is somehow an anti-reason. When talking about the relevance of the Lebesgue Integeral I of course mention a standard example like the Dirchlet function but only to add that to be able to integrate this/such functions is not the point.

The issue is that the clear-cut nature of the reason 'you can integrate this function now' tends to overshadow further reasons one might give. Yet then on second thought this reason is not very compelling because to what end would anybody want to integrate that function specifically in the first place. Thus I consider it of some value to actively stress this point.

First, we have the two "obvious" motivations.

  • Integrating more functions (e.g. the indicator function for the rationals)
  • Making the space of integrable functions complete

Second, Stein and Shakarchi give the following motivations in the introduction to their book Real Analysis (volume 3 of the Princeton Lectures in Analysis series).

  • Fourier Series (establishing the $\ell^2/L^2$ isometry)
  • Interchanging limits and integrals (e.g. the limit of a sequence of continuous functions may not be Riemann integrable)
  • Length of Curves (finding the length of curves that are only rectifiable, not continuously differentiable)
  • Differentiation and Integration (extending the reciprocity to absolutely continuous functions)

You can easily cook up (or look up) interesting examples to illustrate these four ideas.

Stein and Shakarchi also discuss "the problem of measure" (rigorously defining the area or volume of arbitrary sets is hard!), and why the existence of non-measurable sets means we have to deal with $\sigma$-algebras. You ask about integration, but since measure theory and integration are so intimately related, you may also wish to discuss these topics.

  • A compelling reason should be given for caring about defining the length of a curve that is not $C^1$ at all but finitely many points. – KCd Oct 9 at 0:33

When learning a new concept, it might help to point out what's wrong with the old one. In this particular case, a function which cannot be integrated in the Riemann sense on any interval, but where one can assign a value inuitively:

The Dirichlet function is given by $$f(x)=\begin{cases}1, & \mbox{ if } x \in \mathbb{Q}\\0, & \mbox{ if } x \in \mathbb{R}\setminus \mathbb{Q}.\end{cases}$$

Since $\mathbb{Q}$ is dense in $\mathbb {R}$, the Riemann sums do not converge. However (and this is a part where handwavy arguments need to be inserted), this function is only not equal to zero on a countable subset, so its integral should be equal to zero.

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    That was actually what they told me when I take the class. It's a good example for explaing what is the different concept (Dividing the image into parts instead of the domain), but I have the feeling that this is maybe misleading in the sense that no one will ever calculate the integral of the Dirichlet function (since f=0 a.e.). I'm more looking for an example like: "You want to do [task]? Look, since I know the cool Lebesgue integral, I can do this in a few lines and you need a lot of pages with the Riemann integral" – Markus Klein Mar 14 '14 at 17:45
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    I think that this example can still transport one of the main messages of the Lebesgue integral: You can change a function on a set of measure zero, and the integral won't change. This way, I'm computing the integral of the dirichlet function pretty often - as often as I integrate the zero function. You could try to make a huge motivational arc via functional analysis and quantum mechanics. – Roland Mar 14 '14 at 17:54
  • Integrate $1 - f(x)$? – vonbrand Mar 14 '14 at 20:30
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    A function that integrates to zero isn't very interesting... – vonbrand Mar 15 '14 at 13:39
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    @vonbrand I agree with that statement, but I also think that your suggested function is similarly uninteresting. The point I was trying to make is "This is almost the zero function. Why isn't its Riemann integral equal to zero?" – Roland Mar 15 '14 at 14:06

Reiterating points made in the other good answers: first, we want to have a principle that is, approximately, $\lim_n \int f_n = \int \lim_n f_n$. An immediate issue is clarification of in what senses (plural!?!) we might take limits of functions. Pointwise? Uniformly pointwise? These seem only-slightly different, but, as we know by now, are wildly different in their effect. Uniform pointwise is too strong a constraint... we often simply don't have it. While uniform pointwise limits of nice (e.g., continuous) functions are again nice, that's not so strongly true for merely-pointwise limits. (Of course there is considerable landscape in-between.) A related issue is, then, that not-necessarily-uniform pointwise limits of continuous functions need not be continuous... to say the least. So, for the principle of interchanging limit and integral even to have sense, we have to have some compatible (not merely definitional) notion of integral for more general functions... which we want substantially to make sense of these interchange-of-limits theorems.

One side-effect conclusion is that we want descriptions of various classes of functions with metric or other topologies making them complete, in the first place, and then have "integrals" that are continuous linear functionals in that topology.

A meta-point that I find significant is that we really did have some issues needing resolution by extending notions of "integral". Not just trying to extend things for laughs or out of boredom.

E.g., although I was once charmed by the line of patter about integrating the characteristic function of the rationals, we surely can agree that this was not a burning issue needing resolution. It could be made into a more relevant form if we observe that that function is a pointwise limit of continuous functions!

(And, by the way, "the Riemann integral" slightly mis-represents the attitudes of people pre-1850, who knew quite well what they wanted, how to differentiate and integrate. It seems that in their context, the fundamental theorem of calculus, that the derivative of $\int_{x_o}^x f(t)\,dt$ is $f(x)$ for reasonable functions, was the decisive additional feature of integrals... beyond computing areas and such. In particular, they didn't have a "definition" of integral...)

Why not motivate it be being a bit more general, maybe introducing Lebesgue-Stieltjes integration? That way it is easy to see it is very useful in, for instance, probability theory, by giving a unified theory for continuous and discrete probability ( and even finding some new phenomena, continuous distribution not being absolutely continuous, like the "Cantor distribution").

Since this does not seem to be mentioned in other answers, it should be noted that strictly speaking, the Riemann integral is only formulated for functions that are bounded on a compact interval [a,b]. But the Lebesgue integral does not assume compactness of the function domain, nor does it assume the function is bounded.

I think every really convincing argument must make use of the fact that one can take integrals with respect to arbitrary measures. If all one ever does is integrating functions with respect to Lebesgue measure, one is really doing absolute Henstock-Kurzweil integration which can be introduced much like the Riemann integral and actually integrates more functions than the Lebesgue integral (the Henstock-Kurzweil integral is a non-absolute integral).

Given that, the Lebesgue integral can be easily motivated by probability theory. A function on a probability space is measurable (a random variable) exactly when one can meaningfully calculate the probability that the value lies within some interval. If one then wants to calculate the expectation of a bounded random variable, the Lebesgue integral is the obvious correct way to do it. Indeed, many people learned the Lebesgue integral first in the context of probability theory (I'm one of them).

I would not let the desire for motivation lead you to pushing something in a manner that gives the student the wrong insights. The bottom line is that there is almost no important scientific or commercial applications of the L integral (consider that none of the answers to date here on this site, full of very smart people, has listed one). Also consider that most chemists, physicists, engineers go their entire training without bothering to learn it or having it as a required tool.

Just be clear with the students that this is a logical oddity necessary for a more complete course in analysis. I think it will actually all go over better if you are just sort of "man to man" that it is a sort of curiosity of math versus contorting yourself to motivate it as useful (it is not).

"Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane." -Hamming

Hamming's point is more than cute. A modern airplane is a complicated device that requires several sciences and engineering discplines to make all its devices (metallurgy, lift, electronics, heat cycles, radar, petroleum refining, etc.). You could probably write an article like the famous "I, Pencil" about everything that goes into a modern airplane. In doing so, many fields of math would be encountered as needed tools: differential equations and the calculus, complex numbers [key part of EE], algebra, statistics, trigonometry and spherical trigonometry. But where would you find Lebesgue integration? You honestly would just...not.

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    The question began with When students take a course on real analysis, which to me seems to set the stage for the other answers given here. Your answer seems more appropriate for a student asking this in an advanced calculus course or in one of those 1-2 semester undergraduate courses in "advanced mathematics for engineers and physicists". – Dave L Renfro Oct 7 at 8:57
  • Good point, Dave. – guest Oct 7 at 12:18

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