Measure and Integration

Syllabus Details

• Unit I: (Riemann-Stieltjes integral and outer measures) A brief review of Riemann-Stieltjes integral; functions of bounded variation; Jordan outer content.
• Unit II: Lebesgue outer measure; null sets; sigma-algebras, and measure spaces, Borel sigma-algebra in the Euclidean space.
• Unit III: (Measures, construction of measures, and Lebesgue measure) Measures; completion of a measure space; constructing measures from outer measures; Lebesgue measure and its properties; Steinhaus’ theorem and Lebesgue density theorem.
• Unit IV: (Measurable functions and limit theorems) Measurable functions; Borel and Lebesgue measurability; types of convergence for a sequence of measurable functions; Egorov’s theorem and Riesz theorem; approximating measurable functions on the Euclidean space with simple functions, step functions, and continuous functions; Lusin’s theorem.
• Unit V: Monotone convergence theorem; Fatou’s lemma; Lebesgue’s dominated convergence theorem; Lebesgue’s criterion of Riemann integrability; absolutely continuous functions and Fundamental theorem of Calculus. Vitali’s set; Vitali’s covering lemma, differentiability of monotone functions; Lebesgue’s differentiation theorem (proof optional).
• Unit VI: ($L^p$ spaces, and product measures) $L^p$ spaces; Riesz-Fischer theorem at least for $p = 1$; denseness of continuous functions with compact supports in $L^p$ for $1 \le p < \infty$; necessity and sufficiency for convergence in $L^p$; Fubini-Tonelli theorem (proof optional).

Course Learning Objective

1. Interpret the theory of outer measures, sigma-algebras, and measure spaces, and solve elementary problems about them.
2. Interpret the properties of Lebesgue measure, construction of measures, measurable functions, and solve elementary problems about them.
3. Interpret integrable functions on measure spaces, $L^p$ spaces, and product measures, and solve elementary problems about them.
4. Analyze the proofs of results in Measure Theory with a critical mind and use those results and the proof-techniques to analyze and decide, with suitable justification, whether a given statement in Measure Theory is true or false.
5. Interpret the theory and theoretical structure of Measure Theory as a prerequisite for the future study subjects such as Ergodic Theory, Fourier Theory, and Operator Theory.

Recommended Books

1. Apostol, Tom M., Mathematical Analysis, second edition, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1974.
2. Bartle, Robert G., The Elements of Real Analysis, second edition, John Wiley & Sons, New York-London-Sydney, 1976.
3. Royden, H. L., Real Analysis, third edition, Macmillan Publishing Company, New York, 1988.
4. de Barra, G., Measure Theory and Integration, New Age International Pvt. Limited, 1981.