Math 501, Real Analysis, Fall 2008: Syllabus

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The chapter and section numbers refer to the course text: G. Folland, Real Analysis: Modern Techniques and Their Applications.

Date Topics TextAssignments
1 9/3 Important inequalities of Holder and Minkowski Class Notes Assignment due 9/10
Solutions, 1-8
2 9/8 Metric spaces: examples, completeness and contraction mapping Class Notes  
3 9/10 Compactness, Total Boundedness Class Notes and Folland, 0.6 Problems for 9/17
Solutions, 9-18
4 9/15 The Arzela-Ascoli theorem Class notes on Arzela-Ascoli  
5 9/17 Riemann-Stieltjes Integrals Class Notes Problems for 9/24
Solutions
6 9/22 Functions of bounded variation
Class Notes  
79/24 Outline of the Lebesgue approach to integration
Algebras and sigma-algebras
Folland, Chapter 1 Problems for 10/1
Solutions
89/26 Algebras and sigma algebras Folland, Chapter 1  
910/1 Measurable functions Folland, Chapter 2, section 1 Problems for 10/8
Solutions
1010/6 Finitely additive measures, measures,
elementary properites, extension problem
Folland, Chapter 1  
1110/8 Caratheodory's extension theorem and applications Folland, Chapter 1, section 4
Problems for 10/15
Solutions
1210/13 Caratheodory's extension theorem and applications:
Lebesgue measure
Class notes on construction of measures
Folland, Chapter 1  
1310/15 Caratheodory's extension theorem and applications:
Lebesgue measure in d diimensions:
a coin tossing space.
Folland, Chapter 1 Problems for 10/23
Solutions
1410/20 Midterm    
1510/23 Properties of Lebesgue-Stieltjes measures;
Non-measurable sets
Folland, Section 1.5  
1610/27 Integration; definition and Fatou's lemma Folland, Section 2.2  
1710/29 Monotone and dominated convergence theorems Folland, Sections 2.2 and 2.3 Problems for 11/5
Solutions
1811/3 Relation between Riemann and Lebesgue integrals Folland, Sections 2.2 and 2.3  
1911/5 Density of step functions, continuous functions;
modes of convergence
Folland, Sections 2.4 Problems for 11/12
Solutions
2011/10 Convergence in measure and a.e. convergence;
Completelness of L^p;
Change of variables
Folland, Sections 2.4  
2111/12 Change of variables;
Product measure and the Fubini-Tonelli theorem
Folland, Sections 2.5 Problems for 11/19
Solutions
2211/17 Product measure and the Fubini-Tonelli theorem, continued Folland, Sections 2.5  
2311/19 Change of variable in Lebesgue integration;
Signed measures
Folland, Sections 2.6, 2.7, 3.1 Problems for 12/3
Solutions
2412/1 The Radon-Nikodym theorem Folland, Section 3.2  
2512/3 Differentiation of Borel measures w.r.t. Lebesgue measure;
Application to bounded variation functions
Folland, Sections 3.3, 3.4 Problems for 12/10
26-2812/3 Differentiation of Borel measures w.r.t. Lebesgue measure;
Application to bounded variation functions
Folland, Sections 3.3, 3.4, Class notes  
2912/16 FINAL EXAM, 12-3, Hill 425 Info on final (Please Read!)