Measure Theory – IMSc

Por: Swayam . en: ,

Week 1:Introduction and Motivation of Measure theory, Jordan measurability and Jordan content Week 2:Basic properties of Jordan content and connection with Riemann integrals, Motivation and definition of Lebesgue outer measure on R^n Week 3:Properties of Lebesgue outer measure on R^n, Caratheodory extension theorem Week 4:Lebesgue measurability, Vitali and Cantor sets, Boolean and sigma algebras
Week 5:Abstract measure spaces with examples: Borel and Radon measures, Metric outer measures, Lebesgue-Stieljes measures, Hausdorff measures and dimension* (extra content) Week 6:Measurable functions and abstract Lebesgue integration, Monotone convergence theorem, Fatou's lemma, Tonnelli's theorem Week 7:Borel-Cantelli Lemma, Dominated convergence theorem, the space L^1 Week 8:Various modes of convergence and their inter-dependence
Week 9:Riesz representation theorem, examples of measures constructed via RRT Week 10:Product measures and Fubini-Tonnelli theorem Week 11:Hardy-Littlewood Maximal inequality and Lebesgue's differentiation theorem Week 12:Lebesgue's differentiation theorem (continued)