**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. (13 lectures)

Integration: Lebesgue integration, Monotone, Dominated convergence theorems, Fatou’s lemma, modes of convergence, Egoroff’s and Lusin’s theorems, Lp spaces, Radon-Nikodym Theorem. (15 lectures)

Product spaces,** **product σ-algebras and measures, Lebesgue measure on Rn, the Fubini and Tonelli theorems, change of variable. (7 lectures)

Differentiation and integration - functions of bounded variation, absolutely continuous functions, fundamental theorem of calculus for Lebesgue integrals. ( 7 lectures)

**Learning Outcomes:**

The students should be able to

- Explain the construction of the Lebesgue measure on Euclidean space
- Determine questions related to different kinds of convergence
- Identify situations where Fubini and change of variables are relevant and compute integrals using these results.
- Relate some of the concepts they learnt in the Probability course

**Text/Reference Books: **

**Text book:**

- Gerald B. Folland, Real Analysis : Modern Techniques and their Applications, Second Ed., John Wiley & Sons Inc; 1999, ISBN: 978-0471317166
- H. L. Royden, Real Analysis, Pearson publications; Fourth Ed., ISBN:978-93-325-5158-9.
- Inder K. Rana, An Introduction to Measure and Integration (2nd Edition), Narosa Publishing House, New Delhi, 2004, ISBN:978-8173194306

**References :**

- W. Rudin, Real and Complex Analysis, McGraw Hill Education; 3rd edition, ISBN: 978-0070619876
- P. Billingsley, Probability and Measure, John Wiley & Sons Inc; Third Ed., ISBN: 978-81-265-1771-8
- R. G. Bartle, The elements of integration and Lebesgue measure, Wiley Classics Library, John Wiley & Sons Inc., New York
- Donald L. Cohn, Measure theory, Birkhäuser, 2015
- 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, ISBN-10: 3540345132, ISBN-13: 978-3540345138
- Frank Jones, Lebesgue Integration On Euclidean Space, Jones and Bartlett Publishers, Inc; ISBN: 978-0763717087
- S. Kesavan, Measure and Integration, Jainendra K Jain Publishers, ISBN-13 :
****978-9386279774