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?
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
Schools where Lebesgue integration is not available
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
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.
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.
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.
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.
