Instructor: Florent Baudier
Office: Blocker 525J
Office hours: Tuesdays 10:30 a.m.-noon or by appointment
Lectures: TR 2:20 p.m.-3:25 p.m. BLOC 110
Course description: Lebesgue measure and integration theory, differentiation, L_p-spaces, abstract integration, signed measures, Radon-Nikodym theorem, Riesz representation theorem, integration on product spaces Textbook:
Real Analysis (Modern Techniques and Their Applications), Second Edition, Gerald B. Folland Exams Homework   Schedule i

Date of Class	Material covered
Thursday 08/25	general introduction, defects of the Riemann integral
Tuesday 08/30	algebras, sigma-algebras, Borel sigma-algebra
Thursday 09/01	generation of the Borel sigma-algebra, elementary systems, abstract measures
Tuesday 09/06	Homework #1 due ,basic properties of abstract measures and basic examples of measures
Thursday 09/08	uniqueness of measures: pi-systems, Dynkin systems, Dynkin theorem, application to the characterization of measures
Tuesday 09/13	uniqueness of the Lebesgue measure, outer measures, Caratheodory theorem
Thursday 09/15	Homework #2 due , premeasures, completeness of measures, Lebesgue-Stieltjes premeasures on the line
Tuesday 09/20	Lebesgue-Stieltjes premeasures on the line, extending premeasures to measures via outer measures
Thursday 09/22	Homework #3 due , Lebesgue measure on the line: construction and properties, Lebesgue measurable sets and comparison with Borel sets.
Tuesday 09/27	abstract measurable functions, Borel sigma-algebra of the extended real line, extended-real valued measurable functions
Thursday 09/29	Homework #4 due (extended to Monday October 3) ,stability of measurability under elementary operations, approximation by simple functions of nonnegative measurable funtions
Tuesday 10/04	Lebesgue integration of nonnegative simple and measurable functions: definition and basic properties
Thursday 10/06	Homework #5 due, Beppo-Levi monotone onvergence theorem, Fatou Lemma
Tuesday 10/11	Fall break, no classes
Thursday 10/13	MIDTERM, Midterm review problems
Tuesday 10/18	Lebesgue integrable functions, Dominated Convergence Theorem
Thursday 10/20	Homework #6 due , modes of convergence
Tuesday 10/25	product sigma algebras and product measures
Thursday 10/27	Homework #7 due (extended to Monday 10/31 8am) , Fubini-Tonelli theorems
Tuesday 11/01	n-dimensional Lebesgue measure, change of variable
Thursday 11/03	Homework #8 due , signed measures, Hahn decomposition
Tuesday 11/08	mutual singularity, Jordan decomposition, absolute continuity
Thursday 11/10	Homework #9 due , Radon-Nikodym theorem, Lebesgue decomposition
Tuesday 11/15	Basic covering lemma, maximal function, weak Hardy-Littlewood maximal function inequality
Thursday 11/17	Homework #10 due (extended to Tuesday November 22), Vitali covering theorem, Lebesgue differentiation theorem
Tuesday 11/22	continuity and differentiability of monotone functions
Thursday 11/24	Thanksgiving
Tuesday 11/29	functions of bounded variation, Jordan decomposition, absolute continuity
Thursday 12/01	Homework #11 due uniform integrability, fundamental theorem of calculus for the Lebesgue integral
Tuesday 12/06	uniform integrability, fundamental theorem of calculus for the Lebesgue integral
Thursday 12/08	no classes
Wednesday 12/14	Final exam