Analysis

Syllabus Details

• Unit I:(Abstract Integration) Set theoretic notation and terminology, the concept of measurability, simple functions, elementary properties of measures, arithmetic in [0, ∞), integration of positive functions, the role played by sets of measure zero.
• Unit II:(Positive Borel Measures) Vector spaces, topological preliminaries, the Riesz representation theorem, regularity properties of Borel measures, Lebesgue measure, continuity properties of measurable functions.
• Unit III:(L^p-spaces) Convex functions and inequalities, the L^p-spaces, approximation by continuous functions.
• Unit IV:(Complex Measures) Total variation, absolute continuity, consequences of the Radon-Nikodym theorem, bounded linear functionals on L^p, the Riesz representation theorem.
• Unit V:(Product Spaces) Integration on product spaces and Fubini’s theorem.

Course Learning Objective

1. Identify regular positive measures on a locally compact Hausdorff space X as members of the dual of C(X).
2. Apply the monotone and dominated convergence theorems and Fatou’s lemma, and use them with density results to solve problems.
3. Construct σ-algebras on product spaces and verify the hypotheses of the Fubini and Tonelli theorems.
4. Explain the Radon-Nikodym theorem, Lebesgue’s decomposition theorem and the Hahn decomposition theorem, and their applications.
5. Distinguish between the properties of positive, real and complex measures.

Recommended Books

1. Halmos, Paul R. Measure Theory. Springer, 1950
2. Lieb, Elliott H. and Loss, Michael. Analysis. 2nd ed. Graduate Studies in Mathematics. American Mathematical Society, 2001
3. Rudin, Walter. Real and Complex Analysis. Tata McGraw-Hill, 1987