Course information

Introduction: The problem of measure. Classes of sets: Algebras, σ-algebras, Dynkin classes. Measure spaces: Outer measures, complete measures, and completion of a measure space, regular measures, Lebesgue measure. Measurable sets: Structure of measurable sets, non-measurable sets. Measurable functions: Sequences of measurable functions, Egorov and Lusin Theorems. Lebesgue integral: Integral of simple functions, integral of non-negative measurable functions, basic properties of the integral. Monotone convergence theorem, Fatous lemma. Lebesgue integral in general: Dominated convergence theorem, Beppo Levi theorem. Comparison: Riemann and Lebesgue integration. Topics: Modes of convergence of sequences of measurable functions, L p L p spaces. Product measures, Fubini theorem. Signed measures, Radon-Nikodym theorem.