Dissertation on Integrals

Question

I'm considering doing a dissertation on Integrals: Riemann, Henstock-Kurzweil, Lebesgue and more

I'm wondering if I can do it. This is usually a masters level dissertation (while I'm an undergraduate which means I haven't covered as many courses as masters students)

How complex do you think it is? (I'm not looking for an easy option, but I don't want to mess up my first class degree either)

What is expected from me? Obviously I can't come up with new theories myself since I simply don't have the theory behind it. Am I expected to simply understand and rewrite?

Thanks!

Answer 1

My opinion is that this is overly ambitious, but if you decide to go through with it, here are two books that will be very useful:

A Garden of Integrals by Frank Burk

Varieties of Integration by C. Ray Rosentrater

One drawback to these two books is that they make no mention of the large number of other integrals that have been studied. Obviously, the authors have to draw a line somewhere in what they discuss, but a short two or three page appendix or afterword mentioning the integrals of Bochner, Burkill, Denjoy, Jeffery, Khintchine, Kolmogorov, Kubota, Perron, Ridder, Saks, Ward (and others I've probably overlooked) would have been a very useful addition. Indeed, both books treat mostly the same integrals, so their union doesn't tell you about the existence of much more than either of them.

As a partial remedy, there is Gordon's book:

The Integrals of Lebesgue, Denjoy, Perron, and Henstock by Russell A. Gordon

For a more thorough remedy, I recommend these two very extensive survey papers:

Peter Bullen, Non-absolute integrals in the twentieth century, AMS Special Session on Nonabsolute Integration, 23-24 September 2000, 27 pages. (195 references)

Ralph Henstock, A short history of integration theory, Southeast Asian Bulletin of Mathematics 12 #2 (1988), 75-95. (262 references)

However, rather than attempt a survey of integration methods, I recommend focusing on a specific integration topic, such as is discussed in my 7 November 2007 sci.math post and in the math overflow question Cauchy's left endpoint integral (1823).

Another topic is the investigation of what can be the set of all possible Riemann sums for a certain function or for functions having certain specified properties. I know of quite a few papers on this topic, but they're at home now and I don't remember enough about their titles or authors to list any of them now. (One of the authors might be I. J. Maddox.)

Answer 2

Absolutely you should do this, even if it isn't for credit. You'll learn more doing this than you will in two years of classes. Make sure you include plenty of examples and err on the side of too much detail rather than trying to be terse. One thing you might consider is using Riemann to introduce the need for Lebesgue then moving from Lebesgue to Bochner (for functions taking values in Banach spaces) and then to stochastic integration as the most general. I think it is best to have a logical progression rather than cover a bunch of disjoint topics. However, you are more likely to finish (or start) if you are interested in what you are writing about so prioritize that. Also, if you write a quality paper and do quality research, it will open doors in your department beyond your wildest dreams.