TOP TEN REASONS FOR DUMPING THE RIEMANN INTEGRAL #8

What would Sir Isaac say?

Newton died in 1727 so contacting him about Riemann (1826 – 1866) was not easy. But, with some help from a single malt scotch and a ouija board, I have managed to channel him. Here is the session, as best as I can recall.

He first gave me his definition of the integral.

Classical Definition. A function f has an integral on an interval [a,b] provided there is a function F defined on the interval and a subset N of the interval for which

• N is a finite set
• f is defined at all points of [a,b] except possibly points of N.
• F'(x)=f(x) at all points x of [a,b] expect possibly points of N.
• F is continuous at points of N.

In that case the value of the integral is assigned to be F(b)-F(a).

Modern Definition. A function f has an integral on an interval [a,b] provided there is a function F defined on the interval and a subset N of the interval for which

• N is a set of measure zero.
• f is defined at all points of [a,b] except possibly points of N.
• F'(x)=f(x) at all points x of [a,b] expect possibly points of N.
• F has zero variation on N.

In that case the value of the integral is assigned to be F(b)-F(a).

[A brief explanation. Only the notion zero variation on a set may be unfamiliar. You can find it in our dripped analysis text. N is a set of measure zero if the identity function F(x)=x has zero variation on N. The connection between the two definitions is that F has zero variation on a finite set N if and only if F is continuous at each point of N.]

-----------------------transcript of ouija board session---------------------

Me: Is this the correct definition of the integral?

Newton: Certainly. Integration on the real line is antidifferentiation. You can characterize that in numerous other ways, but that is what it is and that should be disclosed to students up front.

Me: Your definition has been called descriptive.

Newton: Indeed. It describes exactly what an integral is. Don't get confused by the fact that there are constructive versions. Start with this and get the constructions later.

Me: Why two versions?

Newton: For the 18th century the ``finite exceptions'' version was adequate for all applications. It was only with Fourier's introduction, in the 19th century, of series representations by trigonometric series that the ``measure zero exceptions" version became needed. Teach the former to your integral calculus students and save the subtle version for later.

Me: You say "get the constructions later." Who did that?

Newton: First there is Cauchy. He clarified the notion of continuity. We had a vague idea before, but he nailed it and showed that continuous functions have integrals. He also characterized the situation for functions with finitely many discontinuities.

Me: And ...

Newton: Certainly Lebesgue. He developed all the tools to characterize, construct the integral, and to investigate the class of absolutely integrable functions [i.e., both f and |f| are integrable].

Me: And ...

Newton: Well that leaves only Denjoy who gave an elaborate treatment for the constructibility of the integral of the nonabsolutely integrable functions.

Me: What about the Riemann integral?

Newton: A minor observation. Also misleading. It pushed integration theory in the wrong
direction for too long. His so-called "integral" only works for bounded functions and derivatives need not be bounded. Also it doesn't handle even bounded derivatives. As soon as that was realized then Lebesgue took it up as a problem: How to integrate constructively all bounded derivatives. He succeeded, of course.


Me: About the Henstock-Kurzweil integral, do you ....

Newton: That's just another descriptive integral. It looks kind of constructive, only because it is so similar to the Riemann integral. Its equivalent to my integral. In fact you can show it is included in my integral in a few short steps. Some people think you could start teaching integration theory with it, but I much prefer my own integral as a starting point. As I said, integration on the real line is antidifferentiation. Start there ...

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[Sorry I don't recall any more, except that he began railing against Leibnitz, Locke, Pepys and some others.]