Module I (10 hours)
Lebesgue outer measure, Measurable sets and Lebesgue measure, Non-measurable sets, Semi-algebras, algebras, monotone class, sigma-algebras, Borel sets, Measurable functions, Simple functions, Littlewood’s three principles.
Module II (10 hours)
Lebesgue integral, Fatou’s lemma, Lebesgue monotone convergence theorem, Lebesgue dominated convergence theorem, Lp spaces, Minkowski's and Holder's inequalities, Completeness of Lp spaces,
Module III (10 hours)
Convergence in measure, Differentiation of monotone functions, Functions of bounded variation, Absolute continuity, Abstract measure spaces, Completion of measure, Product measure, Fubini’s theorem.

To introduce the concept of Lebesgue integral via Lebesgue measure on real line, and extend this idea to abstract spaces and higher dimensional Euclidean spaces.

1. Students will be able to understand the limitations of Riemann integration and the need of extension of the concept of length/volume to measure.
2. They will get acquainted with the Lebesgue integral and related results.
3. They will be able to do manipulations of sums and integral.

H. L. Royden, Real Analysis, Macmillan Publishing Company

G. De Barra, Measure Theory and Integration, Ellis Horwood Publishing Corporation

E. D. Benedetto, Real Analysis: Foundations and Applications, Springer

I. K. Rana, An Introduction to Measure and Integration, Narosa Publishing House