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In this course, you will learn
- 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)
