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 Text Assignments
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
7 9/24 Outline of the Lebesgue approach to integration
Algebras and sigma-algebras Folland, Chapter 1 Problems for 10/1
Solutions
8 9/26 Algebras and sigma algebras Folland, Chapter 1
9 10/1 Measurable functions Folland, Chapter 2, section 1 Problems for 10/8
Solutions
10 10/6 Finitely additive measures, measures,
elementary properites, extension problem Folland, Chapter 1
11 10/8 Caratheodory's extension theorem and applications Folland, Chapter 1, section 4
Problems for 10/15
Solutions
12 10/13 Caratheodory's extension theorem and applications:
Lebesgue measure
Class notes on construction of measures Folland, Chapter 1
13 10/15 Caratheodory's extension theorem and applications:
Lebesgue measure in d diimensions:
a coin tossing space. Folland, Chapter 1 Problems for 10/23
Solutions
14 10/20 Midterm
15 10/23 Properties of Lebesgue-Stieltjes measures;
Non-measurable sets Folland, Section 1.5
16 10/27 Integration; definition and Fatou's lemma Folland, Section 2.2
17 10/29 Monotone and dominated convergence theorems Folland, Sections 2.2 and 2.3 Problems for 11/5
Solutions
18 11/3 Relation between Riemann and Lebesgue integrals Folland, Sections 2.2 and 2.3
19 11/5 Density of step functions, continuous functions;
modes of convergence Folland, Sections 2.4 Problems for 11/12
Solutions
20 11/10 Convergence in measure and a.e. convergence;
Completelness of L^p;
Change of variables Folland, Sections 2.4
21 11/12 Change of variables;
Product measure and the Fubini-Tonelli theorem Folland, Sections 2.5 Problems for 11/19
Solutions
22 11/17 Product measure and the Fubini-Tonelli theorem, continued Folland, Sections 2.5
23 11/19 Change of variable in Lebesgue integration;
Signed measures Folland, Sections 2.6, 2.7, 3.1 Problems for 12/3
Solutions
24 12/1 The Radon-Nikodym theorem Folland, Section 3.2
25 12/3 Differentiation of Borel measures w.r.t. Lebesgue measure;
Application to bounded variation functions Folland, Sections 3.3, 3.4 Problems for 12/10
26-28 12/3 Differentiation of Borel measures w.r.t. Lebesgue measure;
Application to bounded variation functions Folland, Sections 3.3, 3.4, Class notes
29 12/16 FINAL EXAM, 12-3, Hill 425 Info on final (Please Read!)

	Date	Topics	Text	Assignments
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
7	9/24	Outline of the Lebesgue approach to integration Algebras and sigma-algebras	Folland, Chapter 1	Problems for 10/1 Solutions
8	9/26	Algebras and sigma algebras	Folland, Chapter 1
9	10/1	Measurable functions	Folland, Chapter 2, section 1	Problems for 10/8 Solutions
10	10/6	Finitely additive measures, measures, elementary properites, extension problem	Folland, Chapter 1
11	10/8	Caratheodory's extension theorem and applications	Folland, Chapter 1, section 4	Problems for 10/15 Solutions
12	10/13	Caratheodory's extension theorem and applications: Lebesgue measure Class notes on construction of measures	Folland, Chapter 1
13	10/15	Caratheodory's extension theorem and applications: Lebesgue measure in d diimensions: a coin tossing space.	Folland, Chapter 1	Problems for 10/23 Solutions
14	10/20	Midterm
15	10/23	Properties of Lebesgue-Stieltjes measures; Non-measurable sets	Folland, Section 1.5
16	10/27	Integration; definition and Fatou's lemma	Folland, Section 2.2
17	10/29	Monotone and dominated convergence theorems	Folland, Sections 2.2 and 2.3	Problems for 11/5 Solutions
18	11/3	Relation between Riemann and Lebesgue integrals	Folland, Sections 2.2 and 2.3
19	11/5	Density of step functions, continuous functions; modes of convergence	Folland, Sections 2.4	Problems for 11/12 Solutions
20	11/10	Convergence in measure and a.e. convergence; Completelness of L^p; Change of variables	Folland, Sections 2.4
21	11/12	Change of variables; Product measure and the Fubini-Tonelli theorem	Folland, Sections 2.5	Problems for 11/19 Solutions
22	11/17	Product measure and the Fubini-Tonelli theorem, continued	Folland, Sections 2.5
23	11/19	Change of variable in Lebesgue integration; Signed measures	Folland, Sections 2.6, 2.7, 3.1	Problems for 12/3 Solutions
24	12/1	The Radon-Nikodym theorem	Folland, Section 3.2
25	12/3	Differentiation of Borel measures w.r.t. Lebesgue measure; Application to bounded variation functions	Folland, Sections 3.3, 3.4	Problems for 12/10
26-28	12/3	Differentiation of Borel measures w.r.t. Lebesgue measure; Application to bounded variation functions	Folland, Sections 3.3, 3.4, Class notes
29	12/16	FINAL EXAM, 12-3, Hill 425	Info on final (Please Read!)