Syllabus

Main topics

Measures: sigma-algebras, measures, outer measure, Caratheodory extension theorem, Lebesgue and Lebesgue-Stieltjes measures on the real line.
Integration: measurable functions, simple functions, L⁺ and L¹, Lebesgue Monotone and Dominated convergence theorems, modes of convergence, theorems of Egoroff and Lusin, product measures, Tonelli-Fubini theorems.
Signed measures: Hahn decomposition, Jordan decomposition, Lebesgue Radon-Nikodym theorem. Lebesgue Differentiation Theorem.
Intro to Functional Analysis: Normed spaces, linear functionals, L^p spaces (Holder, Minkowski, dual space), Riesz representation theorem, weak convergence, introduction to Hilbert spaces.

Algebras and sigma-algebras of sets. Measures. Measure spaces.
Completeness (for measure spaces). Outer measure.
Premeasure. Caratheodory theorem. Lebesgue measure.
Lebesgue-Stieltjes measures. Examples. Regularity of Lebesgue-Stieltjes measures.
Measurable functions. Simple functions.
Integration of positive functions. Class L^+. Monotone Convergence Theorem.
Fatou's Lemma. Integration of general functions. Basic properties.
Dominated Convergence Theorem. DCT for series. L^1 as a Banach space (including completeness).
Approximation of integrable functions by simple functions, and (for Lebesgue-Stieltjes) by C_c(R).
Lebesgue integral as a generalization of the Riemann integral. Examples.
Modes of convergence and their relationship.
Products of measure spaces. Theorems of Tonelli and Fubini. Regularity and change of variables formulas.
Signed measures, Hahn decomposition, Jordan decomposition
Absolute continuity and mutual singularity of measures. Lebesgue decomposition.
Radon-Nikodym theorem. Conditional expectation. Some examples.
Lebesgue differentiation theorem.
Weierstrass and Stone-Weiertrass theorem.
Linear functionals. Riesz representation Theorem.
L^p space: Holder, Minkowski inequalities; completeness. Riesz: L^q=(L^p)^*.
Intro to Hilbert spaces and Fourier series.