Module I (10 hours)
Lebesgue outer measure, Measurable sets and Lebesgue measure, Non-measurable sets, Semi-algebras, algebras, monotone class, sigma-algebras, Borel sets.

Module II (12 hours)
Measurable functions, Simple functions, Littlewood’s three principles, Lebesgue integral, Fatou’s lemma, Lebesgue monotone convergence theorem, Lebesgue dominated convergence theorem,

Module III (6 hours)
Lp spaces, Minkowski's and Holder's inequalities, Completeness of Lp spaces,

Module IV (12 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 higher dimensional Euclidean spaces and abstract spaces.

CO1. Students will be able to understand the limitations of Riemann integration and the need of extension of the concept of length/volume to measure.

CO2. They will get acquainted with the Lebesgue integral and related results.

CO3. They will be able to do manipulations/interchanges related to sums and integral.

CO4. They will learn the properties of Lp spaces.

CO5. They will learn about different types of convergence and relation between them along with extensions of fundamental results of calculus of one variable.

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