# Should an undergraduate math program contain a course on Lebesgue integration?

Is it standard for a math undergraduate program to have a course on Lebesgue integration?

Does Riemann integral suffice for undergraduates?

The reason of my question is I read a paper by Bartle titled Return to Riemann Integral. In his paper, Bartle claims that most undergraduate US students are not ready to study Lebesgue integration. The paper was written in 1996. Did anything change since then?

• If this math program includes probability theory (which uses the Lebesgue integral for the expected value) or functional analysis ($L^p$ spaces require the Lebesgue integral), then it's a necessary prerequisite. Mar 23 at 20:53
• Note: Bartle's article doesn't advocate for using the standard Riemann integral, but rather its generalization by Kurzweil and Henstock (around 1960). And still, this generalization is only concerned with integrating over $\mathbb{R}^n$ or its subsets. Mar 23 at 21:13
• Most undergraduate US students aren't ready to study even the Riemann Integral. They likely have seen a number of lectures about it, but they didn't really understand what went on. Mar 24 at 1:03
• One of the glitches in this framing of the question is the palpable fact that it's not really the definition of any sort of integral that's needed or used, but its properties. Sure, we like "definitions" that facilitate proving such things, etc., but, after the initial set-up, we rarely return to the "definitions". Just as in basic calculus. Mar 24 at 1:12
• I deleted five comments here where people were arguing about a distraction; if anyone believes a question could be clarified by editing, you can all edit the question to improve it. Also thank you for raising flags when comment threads become unproductive. Mar 26 at 18:18

I think the existing answers understate how much a standard American math major does not see the Lebesgue integral. I'm going to poke around at a variety of college websites to see how they cover this material. Of course, this is not systematic, but I think it is more representative than the other answers. Also, of course, in most cases I only read the school's webpages, so I may be missing something.

I would like to emphasize that, while this list spends most of its effort on elite schools, the majority of undergraduates attend community colleges and the majority of undergraduates pursuing bachelors degrees attend schools with non-competitive admissions. Also, this is an answer about math majors. Most students taught by a math department are not math majors.

Schools where rigorous definitions of integral are not available

• The majority of college students are enrolled in community colleges. Here are the math courses at Washtenaw Community College, the nearest community college to me. It looks to me like none of the calculus courses involve rigorous proofs at all (though Linear Algebra does), and there is no analysis beyond Calculus III.

Schools where Lebesgue integration is not available

• Alright, you may say, but by "math major" I meant "math major at a four year institution pursing a Bachelor's degree." The majority of those are at schools who accept the majority of their applicants. Here is Albion College, chosen because it was the first college in Michigan listed here. Here, it looks like almost every route through the math requirements involves Math 331: Real Analysis which is based on the Riemann integral. I see no course on Lebesgue integration or Measure Theory in their course listings.

Schools where Lebesgue integration is an unusual elective

• Some students are small liberal arts colleges (SLCs). Their math programs vary widely; I'll choose Williams as a college which has a reputation for an excellent math program. The highest mandatory analysis course is Math 350. Looking at the course webpage, I only see Riemann Integration, and the assignments only go up to Chapter 9 in the textbook, which is the construction of the Riemann integral (although the last chapter is the Lebesgue integral). Measure theory is an elective which is not taught every year. So, even at an elite SLC like Williams, Lebesgue integration isn't on the natural track, and it looks like students may have to put in effort to seek it out.

• Many students are at state universities. A mid-ranked state university is Arizona State. It looks like the most advanced mandatory analysis course at Arizona State is Math 371, "Advanced Calculus I", which is based on the Riemann Integral. However, one can choose to study Lebesgue integration in Math 473, "Intermediate Real Analysis II".

• An example of a research-heavy state university (R1) would be Michigan State. At Michigan State, the standard analysis courses for majors are Math 320 and Math 421. I couldn't find a detailed topics list, but their text books are Abbot, "Understanding Analysis" and Wade, "Introduction to Analysis", both of which use the Riemann Integral. There are also honors versions of these courses: Math 327H and Math 429H. Math 327H uses the Riemann Integral; I couldn't find a statement for Math 429H. However, at Michigan State, some students will see the Lebesgue integral, if they take the graduate course Math 828.

Students where I would guess most math majors don't see Lebesgue integrals, but a sizeable minority do

• My own university, the University of Michigan, is what is called a "flagship university", a state university which attempts to have a world class research department. The highest level analysis course which is required of math majors is Math 351: "Principles of Analysis", which uses the Riemann integral. Students bound for graduate school often take the honors version, Math 395/396. Here the Lebesgue integral may or may not be covered by preference of the instructor. When I teach it, I use Riemann integrals; I believe that Prof. Barrett does teach the Lebesgue integral as a side topic but bases most of the course on the Riemann integrals. Students who want to be sure to see the Lebesgue integral should take Math 597, which is targeted primarily at first year grad students but has some undergraduates in it. So students are not required to learn the Lebesgue integral at this level, and may or may not see it naturally in undergraduate courses, but the option definitely exists.

• Back when I was in school, Harvard University was top ranked for math. Harvard math concentrators must take a course in analysis, meaning a course numbered between 112 and 118, and the recommendation of the department is to take either Math 113 "Complex Function Theory" or Math 114 "Analysis of Function Spaces, Measure and Integration". Math 114 teaches Lebesgue integration and 113 doesn't. When I was at Harvard, 113 was much more popular; I don't know if this is still true. So one of the two analysis courses that math concentrators are routed to teaches Lebesgue integration and the other doesn't. On a personal note, I made the very ill-advised decision to take Math 213, the graduate version of Complex Analysis, and was in way over my head. The professor frequently referred to sets of measure zero, so I guess we were using the Lebesgue integral, although I don't recall a specific statement. I have never taken a course which constructed the Lebesgue integral.

Schools where it might be true that the majority of pure math majors see a Lebesgue integral

• MIT is considered one of the most challenging math departments in the nation. Here pure math majors must take 18.100, which uses the Riemann integral, and can then choose between 18.101 (analysis on manifolds), 18.102 (functional analysis) and 18.103 (Fourier analysis). Looking at the webpages, it looks like 18.101 uses Riemann integrals and the others use Lebesgue integrals. There is also an applied math major, which looks like it would not see Lebesgue interals.

• I looked into U Chicago, a school which I think of as having very high standards. Since I don't know this school personally, it is hard to judge what the typical math major does. At Chicago, the highest mandatory analysis course is the 20300/20400/20500 sequence. The textbook, "A textbook for advanced calculus" by Boller and Sally, uses Riemann integrals. (As a sidenote, I had to go to Course Hero to get the syllabus or to see the textbook. Come on, people, make a course webpage!) However, there are two other undergraduate sequences, Math 20700/20800/20900 and Math 31200/31300/31400 which teach the Lebesgue integral. I do not know how popular they are.

Schools where all math majors learn about the Lebesgue integral

• At Caltech, it looks to me like Math 108b is a required course for math majors (or can only be replaced by more advanced courses), and it uses the Lebesgue integral. I don't know Caltech personally and their requirements page is very terse; corrections are welcome.
• I want to emphasize that Albion College is comparatively selective for a four-year college, even if it might be the second least selective (after Washtenaw Community College) on your list. Mar 24 at 18:26
• Most students are at state universities that are not one of the top universities in their state. A good example might be Northern Michigan University. Their math major does not require ANY analysis course (beyond a probably proof-free calculus sequence). Their first analysis course is an elective on a short list of choices; probably most of their math majors (which might only be a handful a year) take it, and it does not cover Lebesgue integration. There is a possibility the second analysis course, which is probably almost never offered, sometimes does. Mar 24 at 19:30
• Thanks. I found nmu.edu/mathandcomputerscience/mathematics-courses , which lists nmu.edu/bulletin/… ; oddly, your link doesn't list it. Mar 24 at 19:50
• In fairness to Williams, I went to another SLC where all of our upper-division courses might meet the same criticism (maybe not the right way to put it?) in this post: everything past diff eq was offered every-other year. So we juniors and seniors took all our classes together, some taking "A year" courses before "B year," others taking "B year" before "A year." Point is: not being offered every year may not be the knock it seems; it still may be on the menu for every student, as it was at my school. (Oh, and +1 anyway for an excellent, thorough answer!) Mar 25 at 2:37
• It was interesting to see Albion College mentioned, as I very rarely see mentioned here (or in MSE, or in mathoverflow) any of the 15-20 colleges I've had an on-campus interview at (4 different search years, over a period of about 10 years). I interviewed there in Spring 1999 (they wound up hiring Bollman), and I recall some pleasant and lengthy discussions with Messer, including looking over a manuscript version of his topology book which for some reason wasn't published until 2006. Mar 25 at 7:41

Is it standard for a math undergraduate program to have a course on Lebesgue integration?

No (assuming that "have a course" means "require people to take such a course in order to get a degree").

But you don't have to take my word for it. Go ahead and check the catalogs of a few top-flight schools in the US.

Does Riemann integral suffice for undergraduates?

Yes.

Michał Miśkiewicz says in a comment:

If this math program includes probability theory (which uses the Lebesgue integral for the expected value) or functional analysis (𝐿𝑝 spaces require the Lebesgue integral), then it's a necessary prerequisite.

No. A normal undergraduate class in probability will simply assume that the functions they're dealing with are sufficiently well behaved that the Riemann integral is the same as the Lebesgue integral. They might make this assumption explicit or leave it implicit. It makes no difference because there are no real-world application in which it matters.

An undergraduate class in probability is mainly populated with stat majors. I can't imagine how an undergraduate statistics program could make an entire course in the Lebesgue measure a prerequisite for their probability course. There's no way their students would satisfy that stack of requirements in time to graduate.

• Thank you for your answer. Could you give an example of a particular program, please? Mar 24 at 6:18
• I'm surprised that it's possible to teach a probability course without talking about random distributions as measures. It's not only the case of whether their densities are Riemann integrable, but whether the densities exist in the first place (for discrete distributions, they don't). But then, it's nice to know the actual state of affairs in the US (+1). Mar 24 at 8:16
• @MichałMiśkiewicz it's not even something covered in some (if not many) masters programs in stats (even really good programs). You'd see it in masters programs that are geared towards being a stepping stone into phd level work, but for most people who are looking to go into industry instead of academia there's no point to that level of depth.
– eps
Mar 24 at 14:04
• Unfortunately, this is true. And many students may go on to graduate school in physics and thus never learn Lebesgue integration. Nevertheless they go ahead and write things with delta functions in there and so on, relying on "intuitive" understanding gleaned from their physics courses. Mar 24 at 17:17
• It makes no difference because there are no real-world application in which it matters. To put it as Richard Hamming said: 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. Mar 24 at 19:10

Is it standard for a math undergraduate program to have a course on Lebesgue integration?

Yes, and I find it bizarre that a university would not have one.

Lebesgue integration (or measure theory more generally) is necessary for a proper grounding in probability theory. Sure, most statisticians won't need it, but it's a very important part of any pure mathematics course. You certainly can't make a rigorous study of probability or statistics without it.

One answer took "have a course" to mean "is a required course". I would not be surprised if an elementary course on Lebesgue integration would be mandatory (although I would certainly not expect a full measure theory course to be mandatory).

Since you are asking about specific programs: I took a measure theory course in third/fourth year that covered Lebesgue integration (among other things). This is now a second-year course called "Lebesgue Measure and Integration". Additionally, if you take Real Analysis in second year then you cover multivariate Darboux/Riemann integration.

The module guide has this to say about the Lebesgue Measure and Integration course:

Lebesgue Measure and Integration:

Essential for: All Pure Analysis modules; Geometry of curves and surfaces.

Useful for: All pure mathematics modules. Dynamical Systems; Statistics modules. Mathematical Finance; Function Spaces and Applications; Advanced Topics in PDEs; modules involving stochastic analysis; Random Dynamical Systems and Ergodic Theory.

MATH50006 Lebesgue Measure and Integration

Brief description: Lebesgue's theory of measure and integration is a powerful extension of the Riemann integral introduced in first-year analysis. It is an essential tool in all aspects of analysis and its applications, including probability, stochastic processes and PDEs.

Learning Outcomes: On successful completion of this module, you will be able to:

• explain key features of the construction of Lebesgue measure, and basic features of the measure as a function,
• explain the limitation of a measure, and measurable sets through algebras of open sets,
• define the measurability of functions in terms of measurable sets, and apply simple criteria to verify measurability,
• outline the Lebesgue integral of a function, and how it relates to the Riemann integral,
• work with spaces of functions as infinite dimensional vector spaces, and algebras, and discuss a variety of such spaces,
• use results on convergence of sequences in measurable functions spaces, and criterial to derive properties of limiting functions and their integrals,
• discuss the relationship between integrability and differentiation,
• explain and use the theorems of Fubini and Tonelli,
• discuss abstract measure theory, and the difficulties with construction of measures and integrals on general spaces.

Module content (syllabus):

An indicative list of sections and topics is:

Motivation: Drawbacks of the Riemann integral, Limits of functions, Length and area, The Fundamental Theorem of Calculus, Measures of sets in R^d,

Measure Theory: Abstract measure theory: Motivation, basic definitions of measure spaces and measures.

Lebesgue Measure in R^d: Volume of rectangles and cubes, The exterior measure, properties, Lebesgue measurable sets, countable additivity. Properties of the Lebesgue measure, Regularity, Invariance, σ-algebras and Borel sets, Non-measurable set.

Measurable functions: Definitions and equivalent formulations of measurability, Sums and products, compositions, limits of measurable functions, “Almost everywhere” properties. Approximation by simple functions and step functions, Egoroff’s and Lusin’s theorems.

Lebesgue integration: Definition using bounded functions on sets of finite measure, Riemann integrable functions are Lebesgue integrable, Integrable functions as a normed vector space, L_p(R^d), dense subsets, Completeness, Fatou’s Lemma, monotone convergence theorem, uniform integrability, Vitali’s Theorem, Fubini’s and Tonelli’s Theorems, statements and proofs.

Differentiation and Integration: Differentiation of the Integral, statement of Lebesgue differentiation theorem, Differentiation of functions, Functions of bounded variation, properties, characterisation, Bounded variation implies differentiable a.e. Absolute continuity of measures, decomposition theorems by Jordan, Hahn and Lebesgue, Radon-Nikodym Theorem.

I can confirm that when I took this course, all of this was covered (along with some extra material about ergodic theory for the mastery component). This is at Imperial College London.

• "I find bizarre [ ...]This is at Imperial College London" You should be used to the fact that the US are bizarre in many aspects, in comparison with the UK :D ! Mar 24 at 18:41
• There's a very similar course at Warwick; there's an answer below pointing out a similar course at Oxford; I found something similar at Cambridge. So absolutely standard in top tier UK maths courses. Mar 26 at 16:45

For undergraduate maths students at the University of Oxford, Riemann Integration is mandatory for 'freshmen' (year 1), and Lebesgue Integration is an elective taken by 'sophmores' (year 2).

Full details of the Lebesgue Integration course, including lecture notes and problem sets, can be found on the Department's public course archive page. The course on Riemann Integration can also be found there.

The elective on Lebesgue Integration is a prerequisite for the junior level courses Functional Analysis I & II, Distribution Theory, and Continuous Martingales & Stochastic Calculus.

Yes. As an elective. If there are interested students we ought to give them a platform to exercise their curiosity. As a requirement, well, that depends on what you aim for the mission of your program. If you want your program to be elite and give students the best chance at hitting graduate school with a well-rounded education then yes, emphatically yes this should be included. The 4-part Princeton Lecture Series on Analysis gives a reasonable path. I found it a worthwhile experience to teach such a course to several gifted students.

• Thank you for your answer. Do you have an example of a particular program where measure theory is an elective? Mar 24 at 6:16
• Apparently, Imperial College London. Mar 24 at 13:43
• In Spain most undergraduate math curricula include measure theory and Lebesgue integration. It is usually part of an elective analysis course and typically is taught in the third year. Apr 5 at 14:16