# 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!)