**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:**

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

**Reference Books:**

- W. Rudin, Real and Complex Analysis, McGraw Hill Education; 3rd edition
- P. Billingsley, Probability and Measure, John Wiley & Sons Inc; Third Ed.
- R. G. Bartle, The elements of integration and Lebesgue measure, Wiley Classics Library, John Wiley & Sons Inc., New York
- 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.
- Terence Tao, An Introduction to Measure Theory, Graduate Studies in Mathematics Vol 126, American Mathematical Society, 2011
- Bogachev, V. I., Measure theory. Vol. I, II. Springer-Verlag, Berlin, 2007
- Frank Jones, Lebesgue Integration On Euclidean Space, Jones and Bartlett Publishers, Inc