How to cure "I don't appreciate Lebesgue integration because I was taught Riemannian integration throughout university"

Question

I was a physics major with applied computation EM background and throughout university I was only taught Riemannian integration.

Every textbook I have read used Riemannian integrals (almost by default assumption). No measure theory were ever introduced.

Only in grad school did I finally learn Lebesgue integration.

But I cannot really appreciate all these beautiful theorems, etc. that can be done using Lebesgue integration because

1. I have to teach other people using textbook written by authors who relies on Riemannian integral by default

2. Many computation software packages seem to have built upon Riemannian integration by default.

3. Too much overhead with Lebesgue integration. I feel like I always have to introduce measure theory (and getting all the miscellaneous technical things) to someone before I can talk about it.

4. At the end of the day, if I were to calculate something, like flux through a plane, or anything involving complex contour integrals, I feel like I have to resort back to Riemannian integration (and the textbooks that base off of it)

Is there anything that can be done in this situation for me to appreciate Lebesgue integration theory?

Answer 1

I think your point 4 is the most important one to start the discussion with. Every mathematical tool has a domain of application which must be considered. The world of deterministic integrals appearing in applied mathematics, physics, engineering, chemistry and the rest of the sciences does not require the ability to have a well-defined integral for pathological functions like the question mark function. Keep in mind the historical context in which the Lebesgue integral was introduced: mathematicians had started realizing that many results which were claimed to have been proven for all functions actually could be contradicted by constructing abstract functions with paradoxical quantities - which had no analogue in the world of science. Thus the purpose of the Lebesgue integral was not to perform new integrals of interest to the sciences, but rather to place the mathematical formalism on a solid foundation. The Lebesgue integral is an antidote to a crisis in the foundations of mathematics, a crisis which was not felt in any of the sciences even as it was upending mathematics at the turn of the 20th century. "If it isn't broken, don't fix it" would be a natural response applied mathematicians and scientists could apply to this situation.

To anyone learning the Lebesgue integral for the purposes of expanding the scope of scientific integration they can perform, I would caution them with the previous paragraph.

However, the power of the Lebesgue integral (and the apparatus of measure theory) lies in its ability to make rigorous mathematical statements that apply to very badly behaved functions - functions that are so badly behaved, they often need to be specifically constructed for this purpose, and have no analogue in "real life". These are functions that are so delicate, an arbitrarily small "tweak" will destroy all these paradoxical properties. (This can be made rigorous in many ways, one of which is the fact that bounded continuous functions are dense in $L^p[0,1]$ - so for any terrible function $f\in L^p[0,1]$ and any $\epsilon>0$ you can find a very nice function $g$ with $|f(x)-g(x)|<\epsilon$ for all $x\in[0,1]$.) In "real life", all measurements carry errors and as a corollary any property that is destroyed by arbitrarily small modifications is not one that can actually be measured!

Despite all this, there is one major application of Lebesgue integration and the measure-theoretic apparatus: stochastic calculus, where one attempts to integrate against stochastic processes like Brownian motion. Even though the sample paths of Brownian motion are continuous, they represent the most badly behaved class of continuous paths possible and require special treatment. While the theory is very well developed in the Brownian case (and many of the top hedge funds on Wall Street have made lots of money exploiting this) there are other stochastic processes whose analysis is much more difficult. (How difficult? Well, the core of the Yang-Mills million dollar problem boils down to finding a way to rigorously define a certain class of very complicated stochastic integrals and show that they have the "obvious" required properties.)

Answer 2

The basic idea behind Lebesgue integration has almost no overhead. When we integrate using the Riemann integral, we cut the graph into vertical strips and ask "how high is each of these strips?" and then add them up. When we integrate using the Lebesgue integral, we cut the graph into horizontal strips and ask "how much of the graph is inside these strips?" and then add them up. For continuous functions, this can be done without measure theory.

The beauty of the Lebesgue integral is that, after investigating how a measure should be defined, we can find the Lebesgue integral of many more functions. This in particular allows us to study problems of the form $$\lim_{n\to\infty} \int_a^b f_n(x)\,dx$$ for a relatively large class of sequences $f_n(x).$ As I understand it, this limit was one of the main reasons why Lebesgue developed the theory in the first place.

Another example: A lot of problems in probability and statistics can be nicely phrased in terms of the Lebesgue integral whereas doing so using the Riemann integral would be cumbersome (if even possible, I've never tried). For instance, when trying to fit a probability model, it is usually appropriate to allow your variable to be the sum of a discrete variable and a continuous variable, which is an easy situation to describe in the context of measure, but as far as I personally know, is all but impossible to describe in the measure-free world (at least, when the dimension of your space is $\geq 2$)

Answer 3

• Every textbook I have read used Riemannian integrals (almost by default assumption).

• Many computation software packages seem to have built upon Riemannian integration by default.

• if I were to calculate something, like flux through a plane, or anything involving complex contour integrals, I feel like I have to resort back to Riemannian integration

I strongly doubt these statements:

• The vast majority of integration in physics employs rules (such as integration by parts) that hold for both Riemann and Lebesgue integration. You hardly ever directly use the Riemann and Lebesgue integral.

• Even if you employ actual Riemann sums to solve an integral, it is very likely of a function that is so well behaving you can easily find some theorem that states that Riemann sums obtain Lebesgue integrals for this class of functions.

• Numerical integration may look a lot like implementing Riemann integration, but it inevitably never goes fully infinitesimal and thus isn’t.

• Some physics textbooks contain introductions to fundamental mathematical concepts like integration (for reasons that are beyond me) and they may use the Riemann integral because they consider it the didactically easiest approach, but if they used any other type of integration, the physics part of the textbook would stay exactly the same.

• Last but not least, there is not a single physical statement (that has been empirically tested) involving an integral that depends on what type of integral is used. The reason for this is that functions that can only be integrated using one type of function simply do not appear in nature, and if they do, we would not be able to tell.

As Pre-Kidney elaborated in more detail, the distinction between Riemann, Lebesgue, and other types of integral is only relevant for inner-mathematical applications, as some type of integral may allow you to better think about certain problems without handling fringe cases. Nature however doesn’t care.

If you feel that you have to decide for one type of integral, I suggest to go for one of the more modern integral types that encompass both the Riemann and the Lebesgue integral, such as the Henstock–Kurzweil integral. I was tutoring a math for physicists course that employed the latter and I haven’t heard of it causing any existential crises, except for being a tad more difficult to grasp than the Riemann integral.