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

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


Text/Reference Books: 

Text book:

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

 

References : 

  1. W. Rudin, Real and Complex Analysis, McGraw Hill Education; 3rd edition, ISBN: 978-0070619876
  2. P. Billingsley, Probability and Measure, John Wiley & Sons Inc; Third Ed., ISBN: 978-81-265-1771-8
  3.  R. G. Bartle, The elements of integration and Lebesgue measure, Wiley Classics Library, John Wiley & Sons Inc., New York
  4. Donald L. Cohn, Measure theory, Birkhäuser, 2015
  5. 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.
  6. Terence Tao, An Introduction to Measure Theory, Graduate Studies in Mathematics Vol 126, American Mathematical Society, 2011
  7. Bogachev, V. I., Measure theory. Vol. I, II. Springer-Verlag, Berlin, 2007, ISBN-10: 3540345132, ISBN-13: 978-3540345138
  8. Frank Jones, Lebesgue Integration On Euclidean Space, Jones and Bartlett Publishers, Inc; ISBN: 978-0763717087
  9. S. Kesavan, Measure and Integration, Jainendra K Jain Publishers, ISBN-13 ‏ : ‎ 978-9386279774