An introductory course in integration and probability

A while ago (in 2002 to be precise), I taught an introductory course of integration, Fourier analysis, and probability in Bordeaux (for third-year university students). While giving the lectures during the first year, I typed them more or less in parallel (in fact, sufficiently close to the course that I could probably have done it also on a weblog, as T. Tao has done this year with his course on Ergodic Theory, and his course on Perelman’s proof of the Poincaré Conjecture…).

Except for using those notes two more years in the same course (or a very similar one the third year), I did not work on them anymore.  But since I had spent a fair amount of time to bring  the text to a reasonable state of polish, and since I’ve found myself using it a few times as a convenient reference for basic facts that I wanted to show to other students, it seems more than reasonable to put this course on the web somewhere.  And so, here it is: Un cours d’intégration.

The text is in French, which diminishes its current interest, but as it is likely that I will teach this topic again in English at ETH, I’ll probably use it as a basis for a translation and/or adaptation.

As the introduction indicates, the only noticeable feature of this integration course (though I don’t think it is at all unique) is that I have treated probability theory in parallel with measure theory, instead of treating probabilistic language and its basic results after the main development, as is often done. I find this is better for a first course, because even students who never go on to another probability course can get, if they are attentive, a good immersion in the special langage and frame of mind of probabilists. (And this was a practical concern in Bordeaux because, typically, there wasn’t another probability course following this one, at that time at least).

The course itself is roughly 140 pages long, and is fairly standard. In measure theory, most of the basic results are treated, though not the Radon-Nykodim theorem. In probability, things go as far as the Central Limit Theorem, but there are no martingales.

The last 40 pages contain the exams given the first two years I gave the course, with corrections.  Some of the problems are dedicated to proofs of quite nice results, including the Radon-Nykodim theorem on [0,1] and Lebesgue’s differentiation theorem, and some are more standard. (Those who have never seen a French-style three/four hours long exam can have a look to get an idea of how this type of things are done in this strange country; with hindsight, it’s clear the exams were quite a bit too difficult for the students I had…)

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Kowalski

I am a professor of mathematics at ETH Zürich since 2008.

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