Top Ten Reasons for Dumping the Riemann Integral #9.

#9. Does the phrase "mildly interesting exercise" suggest anything?


The quoted remark is from Jean Dieudonne, the French mathematician and well known Bourbakiste, in his dismissal of the Riemann integral as a suitable object of study for undergraduate mathematicians.

Actually I find the history itself more than mildly interesting and, with your indulgence, will give a fractured account of it here. What happened to Riemann follows a pattern that I can describe this way:

Suppose that you (Jones) discover an old theorem of Smith that you think you can improve on.

Theorem [Smith] Every object Y has the property Z.

You (Jones) decide that it might be worthwhile to characterize this property Z, especially since Smith seems to have gotten some fame from it. You succeed and write it up like this:

Definition [Jones] We say that any object possessing the property Z is a zamboni.

Theorem [Jones] A necessary and sufficient condition for an object to be
a zamboni is [insert your characterization].

Corollary [Smith] Every object Y is a zamboni.

Thus you (Jones) have, with a stroke of the pen, transformed Smith's theorem into a trivial corollary of Jone's theorem.

The success of Jone's manoever here is whether zambonis are going to have any lasting interest. If they do and prove to be a significant concept, then your fame exceeds Smith's, even though Smith had the original insight. If zambonis fall flat on the mathematical world then move on to something else.

Well, Riemann's zamboni is his integral. He took a theorem of Cauchy asserting that every continuous function on a compact interval [a,b] had a certain property with regard to its integral. He then characterized that property and gave it a name. Hence the definition of the Riemann integral, Riemann's characterization of integrability, and Cauchy's theorem dropping down to the status of a corollary. In fact the sums that Cauchy had used are now called "Riemann sums," so history has taken all the credit from Cauchy and shifted it to Riemann.

Well Riemann doesn't need that credit since he did far better things. Even the Riemann integral was just a throwaway in a 1854 paper about trigonometric series and not anything that he likely spent too much time thinking about. If he had seriously directed his enormous intellect at the problem of integration itself he would certainly have discovered the correct integral.

Unfortunately for Riemann, however, is the fact that this particular zamboni was misguided. Certainly it is a worthwhile professional project to characterize the property that Cauchy had discovered, but it was a sad mistake that future generations employed Riemann's integral as the central tool of integration theory.

For the next fifty years or so (1854--1901) mathematicians took Riemann's integral as if it were the correct one for bounded functions, and spent their time on the problem of integrating unbounded functions. Perhaps they thought of their program as "extending the integral to unbounded functions." Too bad. If you look at the problem this way you fail. If, instead, you tackle the problem of how best to integrate bounded functions, then the unbounded case takes care of itself. Enter the 20th century and Lebesgue. Lebesgue's thesis swept away all the previous theories.

At that point the history gets a bit strange. Lebesgue's theory is considered too difficult for some to learn and for many to teach. So many did not teach it, and many did not learn it. In the last 100 years there have developed many different ways to teach integration theory, some not any more difficult than teaching the Riemann integral itself. But, even so, we still don't teach it until graduate school at many places.