Week 1: Motivation, abstract measures, Caratheodory’s method of extension, completion of a measure.
Week 2: Construction of the Lebesgue meaure, approximation properties.
Week 3: Translation invariance, nonmeasurable sets, measurable functions.
Week 4: Properties of measurable functions, Cantor function.
Week 5: Convergence, Egorov’s theorem, convergence in measure.
Week 6: Lebesgue integration, convergence theorems.
Week 7: Comparison with the Riemann integral, some applications (Weierstrass’ theorem).
Week 8: Differentiation: Monotone functions, functions of bounded variation, absolute continuity.
Week 9: Product spaces, Fubini’s theorem.
Week 10: Signed measures, Radon-Nikodym theorem.
Week 11: L^p-Spaces: Basic properties, approximation, applications.
Week 12: Duality, convolutions.