### Riemann or Lebesgue integration

Excerpt from Martin Davis and Matt Insall, “Mathematics and Design: Yes, But Will it Fly?”, Nexus Network Journal, vol. 4, no. 4 (Autumn 2002).

At 06:39 AM 8/26/2002, Matt Insall wrote:

Personally, I think it is likely to make a difference whether Riemann or Lebesgue integration is used, but that the difference will only be noticeable provided that the designers rigorously adhere to the requirements that that particular integration theory be used for all the integrals. As soon as they revert to numerical approximations, instead of using calculus or analysis, they have muddied the waters so that no direct comparison between one integration theory and another can be made.

Functions that are Riemann integrable are automatically Lebesgue integrable and the values of the integrals will be the same. There are two ways in which a function can be Lebesgue integrable but not Riemann integrable. Any bounded measurable function is L-integrable, but not all of them are R-integrable. Some unbounded measurable functions are also L-integrable, but not Riemann integrable, since all R-integrable functions are bounded. An example: the function 1/sqrt(x) has the L-integral 2/3 on the interval [0,1]. Being unbounded in that interval, it has no R-integral there.

Of course, this function does have an “improper integral” (sometimes called a Riemann-Cauchy integral) in that interval with the very same value. Only functions having absolutely convergent improper integrals have Lebesgue integrals. The classical example is sinx/x on the interval [0,\infnty] which has the value \pi/2 but no Lebesgue integral. This fact is sometimes confusingly stated as a case where something is R-integrable but not L-integrable. But one can perfectly well define “improper”
L-integrals, so that this comparison is inappropriate. It’s just that the definition of L-integration automatically includes cases that in the Riemann case require this extra limiting step for which the inappropriate term “improper” is used.

Hamming of course hadn’t meant to be taken so literally. His aphorism was intended to say that the fine points of mathematical analysis are not relevant to engineering considerations. And, he was perfectly right.

In the long-ago days when I had occasion to be on committees administering an oral qualifying exam for the doctorate, I would often ask the hapless student why analysts prefer the L-integral. This was a trap question: students who fell for the trap would tell me that the function on [0,1] that is 1 on the irrationals and 0 on the rationals is L-integrable, but not R-integrable. The right answer is that the L-integral has useful convergence properties not enjoyed by the R-integral.

Now whatever did Mat Insall have in mind? sin x/x on [0,infty]? Surely not in any engineering analysis.