# The Riemann integral vs Lebesgue integral in several variables for advanced undergraduates

I am about to teach a second course in analysis for advanced undergraduate students. The students have already studied roughly the first eight chapters of Rudin's Principles of mathematical analysis. They have also had a course in Linear Algebra.

I get approximately 36 hours of lectures and the course culminates in the classical theorems of vector analysis. I plan to spend 9-12 hours on differential calculus in $$\mathbb{R}^n$$ and 12-15 hours on vector analysis. This leaves me with around 12 hours of lectures for integration. The question is which integral to cover? It seems impossible to cover the Lebesgue integral in $$\mathbb{R}^n$$ in just twelve hours given that the students have not seen the Lebesgue integral on $$\mathbb{R}$$. On the other hand, looking at chapter 7 of Trench's textbook (http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF), the Riemann integral in several variables seems just as technical as the Lebesgue integral and I am not sure whether the effort has any payoff as the target audience consists of students who are definitely going to pursue mathematics in the future.

My question is whether there is any exposition of the Lebesgue integral in $$\mathbb{R}^n$$ that can be covered in twelve lecture hours. This exposition should contain proofs of both Fubini's theorem and the change of variables. I would also be happy if someone points out the advantages of just sticking to the Riemann integral. It seems to me that it is possible to give a more-or-less complete presentation of the Riemann integral in several variables within twelve lecture hours.

• I feel the idea of measure is difficult to grasp, especially in 1/3rd of a semester. It might be worthwhile looking at a simpler form of Lebesgue integral called Daniell integral - en.wikipedia.org/wiki/Daniell_integral Nov 28 '20 at 10:52
• What would be a good reference for the Daniell integral in several variables that proves the change of variables theorem? Nov 28 '20 at 12:11
• Maybe 7.1-7.8 in Analysis in Euclidean Space (review 1 review 2) by Kenneth Hoffman. 7.1. Motivation (pp. 296-301). 7.2. The Setting (pp. 301-305). 7.3. Sets of Measure Zero (pp. 305-314). 7.4. The Principal Propositions (pp. 314-322). 7.5. Completeness and Continuity (pp. 322-326). 7.6. The Convergence Theorems (pp. 327-336). 7.7. Measurable Functions and Measurable Sets (pp. 336-350). 7.8. Fubini's Theorem (pp. 350-359). (This was the text for a course I took in 1977.) Nov 28 '20 at 15:49
• For another alternative: there exist universities (well, at least one university: Warwick) that use the regulated integral as their introductory integration course. I prefer it in that role, as (a) it dodges some of the technicalities involved in the Riemann integral (by being a little weaker, but ehhh, who actually cares?), and (b) it feels somehow conceptually closer to Lebesque integration (once you get past all of the difficulties of developing measure theory, the definition of the Lebesque integral is mostly just the regulated integral with "step function" replaced by "simple function). Nov 29 '20 at 1:03
• @StevenGubkin Thanks! The book is very good! There is also a more elementary treatment in the book A First Course in Integration by Apsuland and Bungart. Dec 3 '20 at 11:04

Another option (I have never attempted this) would be to claim the existence of a thing called "The Lebesgue integral", list some carefully chosen theorems about it as axioms (maybe just linearity, some inequalities, and dominated convergence), and treat it as a black box for your course. You could give an extremely high level sketch of how this would be defined, and promise them that they could learn the construction in another course. Give them the course number of the course at this university where measure theory is developed in detail for further questions. Whether such an option would be suitable depends on your goals for this course.

The word covered is unproductive, as it is so teacher-oriented. Any amount of material can be covered in 12 hours, provided that you can speak and write fast enough, and when you reach your limit, you prepare slides full with formulas and flip them faster than anyone can even read. The material is covered, right?

If you want a streamlined presentation, take a look at at Folland's "Real analysis: modern techniques and applications". It covers the required material on pages from 19-81, so 62 pages in total. So, you'd need to cover about 5 pages per lecture hour. In my experience, anything faster than 3 pages/hour is unreasonable. And the text is terse and it doesn't get any more streamlined than that. But there are really not so many innovations in Folland, I feel that the material is standard and other sources cover it in essentially the same way, just add more details.

To put it differently, in my university there is a separate 5cr. course (28 hours of lectures) called "Measure and integration", that is supposed to cover approximately first 60 pages of Folland (not including Fubini). And it does not have a reputation of the easiest of courses. Admittedly, course does feel slow to me, and our students are not very strong, but then again, it is impossible to answer your question without knowing how strong your students are.

I do share the sentiment that teaching Riemann integration instead of Lebesgue is feels like a waste of time. But it is a fact that Lebesgue theory takes a non-trivial time to learn.

• I agree with your point about the word "covered". I only meant that it is not possible to teach the Lebesgue integral in way that the average student has a reasonable understanding in just 12 hours. I can definitely "cover" the Lebesgue integral in 12 hours though I doubt whether the students would gain much. To get a gauge of how strong the students are, I believe they would be able to follow the 62 pages from Folland if it was lectured in 24 hours rather than 12. I was hoping to get a good reference for the Lebesgue integral that does not define the integral using measure. Nov 28 '20 at 12:05
• @Jaikrishnan, but how would it be possible? Once you have defined Lebesgue integral, you have the measure for free, by restricting to indicator sets. So, at best you might be able to define the integral without measure, but proving its basic properties cannot be easier than constructing the Lebesgue measure. Nov 28 '20 at 12:27
• Of course, once you have the integral, you have the measure for free. But I am completely uninterested in the measure. I also get to bypass what I find the most unintuitive part of the measure theory approach to the integral: the Caratheodary extension theorem. The recent textbook by Barry Simon, for instance, takes this approach but it would take too much time. The classical book by Apostol also introduces the integral directly. Let me go through Apostol's book and see if the treatment is good. Somehow I was under the impression that Apostol's treatment requires the Riemann integral. Nov 28 '20 at 13:37

The discussion seems to overlook this simple fact: the Lebesgue integral is not a generalization of and cannot serve as a substitute for the Riemann integral.

• I think that this is not relevant to the question here - however, it seem like you mean that the Lebesgue integral is not a generalization of the improper Riemann integral. For the usual Riemann integral (i.e. the one on compact intervals), the Lebesgue integral is a proper generalization. No one forbids to define an "improper Lebesgue integral" in the way one does for the Riemann integral. But usually this is not done, but it's not an inherent limitation…
– Dirk
Dec 2 '20 at 7:01
• @Dirk It's not about improper integration. It's about orientation, to begin with. en.wikipedia.org/wiki/… Dec 2 '20 at 22:57