MTH306: (Lebesgue Measure and Integration Theory)

        [Cr:4, Lc:3, Tt:1, Lb:0]

Course Outline

• Construction of Lebesgue measure, Measurable functions, Lebesgue integration.
• Monotone convergence theorem, dominated convergence theorem, Fatou’s lemma
• Comparison of Riemann integration and Lebesgue integration
• Lp(Rn) spaces (completeness, dense subsets, notion of separability)
• Product sigma algebras, Product measures, Fubini’s theorem in Rn(at least statement with examples)
• Characterisation of continuous linear functionals on Lp spaces.
• Functions of bounded variation, Absolutely continuous functions, Fundamental theorem of Calculus

Additional Topics

bstract measures and integration, signed measure, Radon Nikodym theorem, complex measures, Riesz representation theorem

Reference Texts:

1. Stein and Shakarchi: Real analysis (Measure theory and integration.
2. W. Rudin: Real and Complex Analysis
3. G. B. Folland: Real Analysis, Modem techniques and their applications
4. H.L. Royden: Real Analysis
5. R. Wheeden and A. Zygmund: Measure and Integral: An introduction to Real Analysis.