MA5340 Measure and Integration
Unit I. Riemann integral to Abstract measure [14 lectures] Function theoretic view of Riemann integral, Outer measure induced by the length, the Caratheodary condition, Lebesgue measurable sets in R, Non-measurable sets in R, Abstract Measurable space, Borel σalgebra,Measure, Continuity properties of a measure, Monotone class theorem, Uniqueness of the extension, Completion of a measure space, Completeness of Lebesgue σ-algebra.
Unit II. Measurable functions, convergence and integration [21 lectures] Measurable functions, Convergence of measurable functions (almost everywhere, in measure, in mean, almost uniform), Egorov’s theorem, Lusin’s theorem, Integral of nonnegative measurable function, Monotone convergence theorem, Fatou’s lemma, Lebesgue Integrable functions, Dominated convergence theorem, Generalized dominated convergence theorem, Scheffe’s lemma, Completeness of L’(μ), L1 [a, b] as the completion of R[a, b], Bounded variation, Absolute continuity, Fundamental theorem for Lebesgue integrable functions.
Unit III. Product measure [5 lectures] Product measure, Fubini’s theorem.
1. G. de Barra, Measure and Integration, Wiley Eastern, 1981.
2. I. K. Rana, An Introduction to Measure and Integration, Second Edition, Narosa, 2005.
1. H. L. Royden, Real Analysis, Third edition, Prentice-Hall of India, 1995.
2. Terence Tao, An Introduction to Measure Theory, Graduate Studies in Mathematics, AMS, 2011
3. G. Folland, Real Analysis: Modern Techniques and Their Applications.
4. W. Rudin, Real and Complex Analysis, Third edition, McGraw-Hill, International Editions, 1987