Measure theory

This is an introductory Measure Theory course at the PhD level. A basic outline of the syllabus for the course can be found at this link under Math 813 .

Schedule: The class meets twice a week, Tuesday/Friday 2 - 3.30 pm.

Notes

References:

There is no one book that will be linearly followed. However, most of the material covered in class should be found in some guise in one of the following standard references:

(A) M. E. Taylor, Measure theory and integration. Graduate Studies in Mathematics, 76. American Mathematical Society, Providence, RI, 2006. xiv+319 pp.

(B) E. H. Lieb and M. Loss, Analysis. Second edition. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001. xxii+346 pp.

(C) G. Folland, Real analysis. Modern techniques and their applications. Second edition. Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999. xvi+386 pp.

Grading/Exam: Midsems will be based on presentations (topics to be announced) by students and shall carry 40% of the total credit. Final exam is 60% and the mode will be oral interviews (as decided by popular vote in class).

Proposed topics for midsem presentation: Banach-Tarski (Amanpreet), Whitney extension theorem, Riesz-Thorin interpolation theorem (Soumyadeb), Marcinkiewicz interpolation theorem (Hamidul), Gauss-Green formula on Lipschitz domains, Layer cake representation and bathtub principle (Sayed), Sard's theorem (Amal), Measures on manifolds and change of variables formula, Basics of Sobolev spaces (Rishi), Equivalence of Sobolev and isoperimetric inequalities (Aratrika), Alexandrov's theorem on convex functions (Shirish), Integration in polar coordinates (Bittu), Fourier analysis of measures, Spectral measures and decomposition of spectra (up to the statement of the spectral theorem) (Adarsh).

Contact information:

Email: mathmukherjee@gmail.com or mukherjee@math.iitb.ac.in

Office: G3 -E, Math Department (ground floor from the corridor entrance, first floor from the front entrance)

General advice: It is highly recommended that you consult other standard texts in the field, to appreciate different tastes in selection of materials, or differing slants on the same material.