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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!

Cameron Williams
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GRS
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    It will be quite an undertaking and could easily take an entire year of hard work to do. You should consult with a professor about this. Measure theory is tricky and often takes a few cycles to really understand well. The Henstock-Kurzweil integral is a bit strange in some ways as well but it feels natural in some ways. In undergraduate and master's theses, it is almost always the case that no new results are established, maybe a few new examples, but that's about it. You will be repackaging the work other people have done and providing your own lens to help others understand. – Cameron Williams May 19 '16 at 17:39
  • Any student (as far as I know) who will write a project/thesis/dissertation (it is not clear whether this is some kind of undergrad project or master's thesis) are paired with at least one advisor, and they usually take part in the topic-choosing process. – pjs36 May 19 '16 at 17:49
  • I had a meeting with a professor who wanted to refresh his memory of some integrals and learn another one (I assumed he doesn't know these integrals himself for a fact). He also said measure theory is tricky, but I have a few probability and analysis courses to do next year. I was hoping to manage to understand all these integrals during the summer, so I can complete the thesis by december. I'm really not sure if I should take on it or not – GRS May 19 '16 at 17:52
  • Most of what yo ask you should ask an intructor in your department, really. «What is expected from me?», for example, really depends on what your department wants, and the ones they know is them (we don't even know what department it is!) – Mariano Suárez-Álvarez May 19 '16 at 18:25
  • @MarianoSuárez-Alvarez it's the department of mathematics, my tutor teaches stochastic/finance/probability but he wants an integral project (he said that some of the integrals he doesn't know) – GRS May 19 '16 at 18:32
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    By "department" I meant what university and so on. In fact, we don't even know in what continent you are! (and that assuming that you are on Earth :-) ) You must understand that requirements and "what is expected from me" vary enormously. This is really not the sort of thing one discusses with random people on the Internet. – Mariano Suárez-Álvarez May 19 '16 at 18:35

2 Answers2

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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.)

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Dave L. Renfro
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  • Thanks, I've seen the garden of integrals book, which was also recommended by my professor to read and study. Do you think I could manage time-wise? – GRS May 20 '16 at 00:03
  • A more direct link to Bullen's paper: https://www.emis.de/proceedings/Toronto2000/papers/bullen.pdf – Hans Lundmark May 20 '16 at 06:37
  • @GRS: I think reading though the book, but NOT trying to write a thesis is possible, and I'd probably recommend this unless you have to write a thesis. In fact, since your professor wants to learn some of this material also, perhaps a way to handle your situation would be for YOU to give lectures going through the book (or at least work problems from the book) in front of your advisor (and possibly one or two or more interested other students and/or faculty) a couple of afternoon hours a week. As I'm sure you've heard many times, one tends to learn something especially well when teaching it. – Dave L. Renfro May 20 '16 at 14:02
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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.

Wavelet
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  • Thanks, I have absolutely no theory about it just yet. Do you think I can do it? (I'm mainly taking analysis and probability+stats courses), I also have only half a year (before december) to submit the whole thing. I'm going to work over the summer, ideally I'd have most of it done during the summer, but I might be a bit optimistic. It's the time that I'm worried about, since I will have 7 more courses alongside this thesis. This means I can either do 9 courses, or 7+thesis. – GRS May 19 '16 at 18:30
  • Well that largely depends on the rate at which you absorb new material and your capacity to be productive. If you know absolutely nothing about measure theory today, I'd err on the side of that timeline being a stretch. Just a survey of the Lebesgue integral itself might be possible if you crack open a book today and devote 3 or so hours a day to learning and plan to start writing at the end of the summer but even then, I think the average undergrad would find it difficult to defend such a thesis/dissertation just because of the sheer volume of material and the level of abstraction. – Wavelet May 19 '16 at 18:39