Course Details:

Classes of sets and measures: Sigma algebras, Borel sigma algebra, measure and its properties (monotonicity, continuity etc.), Carathéodory's extension theorem and construction of Lebesgue measure on the real line.

Integration: Lebesgue integration, Monotone, Dominated convergence theorems, Fatou’s lemma, modes of convergence, Egoroff’s and Lusin’s theorems,  Lp space - definition and examples.

Product spaces, product σ-algebras and measures, Lebesgue measure on R^n , the Fubini and Tonelli theorems, change of variable.

Absolute continuity, Radon-Nikodym theorem, Lebesgue decomposition, signed and complex measures, Hahn-Jordan decomposition theorems, Riesz representation theorem for C(K).

Differentiation and integration - functions of bounded variation, absolutely continuous functions, fundamental theorem of calculus for Lebesgue integrals.

Text Books:

  1. Gerald B. Folland, Real Analysis : Modern Techniques and their Applications, Second Ed., John Wiley & Sons Inc; 1999
  2. H. L. Royden, Real Analysis, Pearson publications; Fourth Ed.
  3. Inder K. Rana, An Introduction to Measure and Integration (2nd Edition), Narosa Publishing House, New Delhi, 2004

Reference Books:

  1. W. Rudin, Real and Complex Analysis, McGraw Hill Education; 3rd edition
  2. P. Billingsley, Probability and Measure, John Wiley & Sons Inc; Third Ed.
  3. R. G. Bartle, The elements of integration and Lebesgue measure, Wiley Classics Library, John Wiley & Sons Inc., New York
  4. Elias M. Stein and Rami Shakarchi, Real analysis, Princeton Lectures in Analysis, vol. 3, Princeton University Press, Princeton, NJ, 2005, Measure theory, integration, and Hilbert spaces.
  5. Terence Tao, An Introduction to Measure Theory, Graduate Studies in Mathematics Vol 126, American Mathematical Society, 2011
  6. Bogachev, V. I., Measure theory. Vol. I, II. Springer-Verlag, Berlin, 2007
  7. Frank Jones, Lebesgue Integration On Euclidean Space, Jones and Bartlett Publishers, Inc