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Manjunath Krishnapur

 Department of Mathematics, Indian Institute of Science, Bangalore 560 012

Probability theory (Spring 2015)

Classes: Tue, Thu, 5:40-7:10, LH-1, Mathematics department
Teaching assistant: Kartick Adhikari
Description: This is a first course in measure theoretical probability theory. It is optional for everyone, but undergraduates, int-phd and phd students are all welcome to credit or audit. The prerequisites are basic real analysis, at least a first course in discrete probability. Measure theory is required for the course, but it is possible to learn it concurrently. If you are planning to do that, please know that it requires you to work hard and make extra effort. In the first five weeks, I shall cover various aspects of measure theory, but that is not a substitute for a full course.

Grading: There will be a final exam (50%) and two mid-term exams (total 30%). Many problems will be given and homeworks will carry at least 20% of the final score.

Texts and other resources: I shall put my lecture notes up on the course page (but not always in a timely fashion). There are many good books on the subject, some of which I mention below.
  1. Rick Durrett Probability: theory and examples. Click for a copy!
  2. Patrick Billingsley Probability and measure, 3rd ed. Wiley India. Available in Tata book house
  3. Richard Dudley Real analysis and probability, Cambridge university press
  4. Leo Breiman Probability, SIAM: Society for Industrial and Applied Mathematics
Some comments: Durrett's book is wonderful, but it is a graduate level text and some of the students may find it a little difficult. Billingsley's book is well-written, very detailed and has elaborate explanations. Dudley's book appears to be extremely good and reasonably concise. It has both measure theory and probability theory and hence most suitable for this course. What it lacks are interesting examples of interest in modern probability theory (which is one of the special attractions of Durrett's book).
Notes and homeworks/links:
Other announcements/links:
  • A sweeping view of probability by the greatest probabilist of all time (see also the letter by Aleksandrov at the end). If the link does not work for you, look for the article Probability theory by A.N. Kolmogorov [252–284, Izdat. Akad. Nauk SSSR, Moscow, 1956].
  • A very well-written article on the history of probability up to Kolmogorov's axiomatic definition.
  • We constructed a Borel isomorphism between (0,1)^d and a Borel subset of (0,1). In Dudley's book (appendix C) you will find a bijection between R^d and R using ideas like in the Schröder-Bernstein theorem. Soumyo Biswas pointed out a way to get a bijection by modifying the correspondence using binary digits. The link only discusses it as a bijection. Showing Borel measurability of the map and its inverse is an exercise!