Measure theory is a must

[This post started out as an introduction to a post I was planning to write on convergence theorems for the Riemann integral. The introduction kind of got out hand, so I decided to post it separately. Since I have to get back to my real work, I will postpone writing that post on convergence theorems for the Riemann integral for another time, probably during the Passover break (but in any case before we need them for the course I am teaching this term, Calculus 2)].

Mathematicians love to argue about subjective opinions. One of the most tiresome and depressing subjects of debate is “What should an undergraduate math major curriculum contain?”

Almost everybody agrees with the first part of the following sentence: “One cannot grant a University Degree in Mathematics to someone who does not know what is …”

The ending of this sentence, on the other hand, is a matter of terrible debate. (And when I write that the debate is “terrible” I am not being poetic and I am not exaggerating – but this a different story that I will not go into in public). One may indeed rightfully claim that it is ridiculous that a student will complete a major in math without knowing Number Theory/Measure Theory/ Galois Theory or what is a group action. On the other hand, it would be absurd if our majors don’t even know what is Cholesky decomposition, numerical quadrature, or the P =? NP problem. But these important topics should not come at the cost of Topology, Harmonic Analysis, Complex Analysis, PDEs, or Ring Theory. Etc. etc. etc…

However one tries to plan the curriculum, some topics will be left out. My personal opinion is that it does not matter so much what the undergraduate curriculum contains, so long as it is relatively broad and taught at a high level. The goal is to have majors that are capable of learning more math, either by themselves or in graduate studies.

One claim that I hear often is that one cannot study Functional Analysis or Harmonic Analysis without learning Measure Theory first, or at least concurrently. As someone who has taught functional analysis, operator theory and harmonic analysis a few times, I certainly see the usefulness of knowing some measure theory before hand. It is very convenient for me as a lecturer, especially if I want to use a standard textbook.

However, this claim, although it has some truth in it, is mostly false (it is opposed to experience and to mathematical logic), and is based solely on the personal experience of the claimer (“that’s how I learned it”).

One of the two main reasons that people think that you cannot do functional analysis without measure theory, is that without measure theory you do not have the interesting examples of $L^p$ spaces. In partiulcar the space $L^2(X,\mu)$, precious to operator theorists, is out of the game. But there are two ways solve this problem.

First of all, there are many other interesting examples, such as the classical $\ell^p$ spaces, or many interesting finite dimensional examples. One may object that these spaces are unnatural, and I will not completely disagree. But if you want a natural example of a Hilbert space, for example, then try looking at Hilbert function spaces (see these notes). Arguably, the Hardy space $H^2$ is as natural as you can hope that an infinite dimensional space will be, not one bit less natural than $L^2[0,1]$.

Another solution is that one does not have to give up $L^p$ spaces at all! One just defines them as completions of spaces of continuous functions with respect to the appropriate norm. Of course, the price to pay is that you only have $L^p(K)$ where $K$ is a sufficiently nice subset of $\mathbb{R}^d$, instead of the “more general” $L^p(X,\mu)$. See these notes of mine (Section 4), or better, see this paper of Lax (sorry for putting a link to a paywall – suggestions are welcome).

I have to admit: to obtain the duality theorems such as $(L^p)^* = L^q$ – and thus to open the door to some of the most important applications of functional analysis – one does need to develop some measure theory (at least I don’t know of a slicker way). I did not suggest to abandon measure theory completely. My case was just that one could go a very long way without it, especially if one is concentrating on operators on Hilbert spaces (for $(L^2[0,1])^* = L^2[0,1]$ you don’t need to know what a measure is).

(By the way – in my opinion an undergraduate curriculum that contains no functional analysis is also no big disaster).

The second main reason that people bring to support the claim that measure theory is a must, is that it is a superior theory to Riemannian integration because it has nice closure properties and powerful convergence theorems: The Dominated Convergence Theorem and the Monotone Convergence Theorem of Lebesgue (see the introduction to this Wikipedia article). In particular, knowing these convergence theorems greatly facilitates the study of analytic topics such as Harmonic Analysis or PDEs.

True.

But…

Take for example the DCT. It states that if $f_n$ are in $L^1$ and $f_n \rightarrow f$ almost everywhere, and if there exists $g \in L^1$ such that $|f_n| \leq g$ for all $n$, then $f$ is also in $L^1$ and

$\lim_n \int f_n = \int f$.

Such a theorem cannot hold in the context of Riemann integration, even if one replaces convergence almost everywhere with pointwise convergence everywhere, since the pointwise limit of a sequence of Riemann integrable functions need not be Riemann integrable.

However, in a typical application of this theorem to a computation, the limit function $f$ is evidently Riemann integrable. As it turns out, this is the only thing that goes wrong with the DCT for Riemann integrable functions. To be precise, there is DCT appropriate for Riemann integrals (you just throw in the assumption that the limit function is also Riemann integrable – an assumption that holds in practice many times), that can be taught to first year math majors. This fact is well known – see this paper for a proof and a historical account. Mostly for my own purposes, I hope to write more on this soon.

To sum up, measure theory is indeed a must, but so are many other subjects. So it is good to think how to maneuver when your students did not learn it. Much can still be done!

(Another case for measure theory is that you need it in order to do probability right. That’s completely true. On the other hand, I think that there is nothing wrong with doing things wrong, and probability is an excellent example.)