Manjunath Krishnapur

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

Measure and integration (MA 222) (Jan-Apr 2018)

Tue, Thu 2:00-3:30, LH-4, Mathematics department

Tutorials by Sumit Mohanty: Day TBA, Time TBA LH-1, Mathematics department.

Description: This is a course in measure theory. The official syllabus reads:
Construction of Lebesgue measure, Measurable functions, Lebesgue integration, Abstract measure and abstract integration, Monotone convergence theorem, Dominated convergence theorem, Fatou’s lemma, Comparison of Riemann integration and Lebesgue integration, Product sigma algebras, Product measures, Sections of measurable functions, Fubini’s theorem, Signed measures and Randon-Nikodym theorem, Lp-spaces, Characterization of continuous linear functionals on Lp - spaces, Change of variables, Complex measures, Riesz representation theorem.
The actual coverage of topics may differ slightly due to constraints of time and taste.

Grading: The final grade will be based on homeworks (may be) and midterms (total of 50%) and the final exam (50%). Solving problems (preferably many more than given in the problem sets) is absolutely crucial to develop an understanding of the subject.

Texts and other resources: There are many books on the subject. The following appear to be well-written or interesting in other ways.
  1. Elias M Stein and Rami Shakarchi. Real Analysis: Measure Theory, Integration, and Hilbert Spaces. Princeton University Press (2005).
  2. Terence Tao. An Introduction to Measure Theory. American Mathematical Society. Graduate Studies in Mathematics Volume: 126; 2011.
  3. Walter Rudin. Real and Complex Analysis.
  4. R. M. Dudley. Real analysis and probability. Cambridge University Press; 2nd edition (2002).
  5. Vladimir Bogachev. Measure theory. Springer-Verlag Berlin Heidelberg (2007).
  6. Donald Cohn. Measure theory. 2nd edition. Birkhauser (2013).
The first book is written with amazing clarity. So is the second but much shorter. Dudley's book (first half of it) covers more things. Many other books, such as the one by Folland, one by Royden, etc., are also good as text books. Rudin's book is extraordinary in its contents, but may or may not be the most suitable one to learn from for a first-timer. Bogachev's book is very extensive (if not enough for you, take a look at the five-volume series by Fremlin).
A list of problems