TR 2:20 p.m.-3:25 p.m. BLOC 110
-spaces, abstract integration, signed measures,
Radon-Nikodym theorem, Riesz representation theorem, integration on product spaces
Real Analysis (Modern Techniques and Their Applications), Second Edition, Gerald B. Folland
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 |
i
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 |