Unit 1

Lebesgue Outer Measure, Measurable sets, Regularity, Measurable functions, Borel and Lebesgue Measurability. (2.1-2.5 of Text)

Unit 2

Integration of Non-negative functions, The General Integral, Integration of Series, Riemann and Lebesgue Integrals, The Four Derivatives, Lebesgue’s Differentiation Theorem, Differentiations and Integration. (3.1 to 3.4, 4.1, 4.4 (statements only),4.5 of Text)

Unit 3

Abstract Measure Spaces: Measures and Outer Measures, Extension of a measure, Uniqueness of the Extension, Completion of the Measure, Measure spaces, Integration with respect to a Measure (5.1- 5.6 of Text)

Unit 4

The Lp Spaces, Convex Functions, Jensen’s Inequality, The Inequalities of Holder and Minkowski, Completeness of Lp (μ). (6.1-6.5 of Text)

Unit 5

Convergence in Measure, Signed Measures and the Hahn Decomposition, The Jordan Decomposition, The Radon-Nikodym Theorem, Some Applications of the Radon-Nikodym Theorem. (7.1, 8.1-8.4 of Text)

Course outcomes
CO1: Demonstrate capacity for mathematical reasoning through analyzing, proving and explaining concepts.
CO2: Introduces the notion of a sigma algebra, introduces measurable functions, measures, and examine their properties..
CO3: Study in detail the properties of the Lebesgue integral, and fundamental convergence theorems in Measure and integration namely Lebesgue’s Monotone Convergence Theorem and Lebesgue’s dominated Convergence Theorems.
CO4: In Lp spaces, study in detail about the fundamental inequalities namely Holders and Minkowski’s and hence derive the important fact that spaces. Lp spaces are complete metric
CO5: Introduce the total variation of a complex measure, positive and negative variations of a real measure, and then construct Lebesgue Radon Nikodym theorem which has important applications in Probability theory
CO6: Given two measurable spaces and measures on them , obtain a product measurable space and a product measure on this space.

Textbook

1. De Barra G, Measure Theory and Integration, New Age Publishers.

References

1. Elias M Stein, Real Analysis: Measure Theory, Integration and Hilbert Spaces, New Age Publishers.
2. H.L. Royden and P.M. Fitzpatrick, Real Analysis, PHI.