Office Hours: M: 3-4, W: 1-2, F: 4-5.
Office: 4064 East hall

First Exam: Feb. 21th, 2:10-3pm, 455 Dennison exam1 solution
Second Exam: take home exam (Due April 16) ,
in class exam: April 21th, 2:10-3pm, SEB2229

Week 1:

Basic set theory: cardinals. See Folland: Real Analysis, Chapter 0, and
terrytao.wordpress.com/2009/11/05/the-no-self-defeating-object-argument

Week 2:

cardinals, open and closed sets in $R^n$. Borel sets. See Royden: Chapter 1 and Chapter 2.

Lecture notes , Problems (Due Jan. 22) , solution

Week 3: Tao: an introduction to measure theory, Chapter 1: 1.1, 1.2

Lecture notes , Homework: Exercises 1.1.4, 1.1.5, 1.1.6, 1.1.12, 1.1.16, 1.1.25 (Due Jan. 29) solution (optional hw: 1.1.3*, 1.1.15*, 1.1.18*)

Week 4: Tao: an introduction to measure theory, Chapter 1: 1.2, 1.3, Royden, Chapter 3

Lecture notes, Lecture notes , Lecture notes , Homework: Exercises 1.2.7, 1.2.9, 1.2.14, 1.2.15, 1.2.17,1.2.20, Additional Problems (Due Feb. 5) , solution

Week 5: Measurable functions, modes of convergence: Tao: Chapter 1: 1.3.2. 1.5, Royden: Chapter 3: 3.5, 3.6.

Lecture notes , Lecture notes , Lecture notes , Problems (Due Feb. 12) , solution

Week 6: Littlewood's three principle, modes of convergence, Lebesgue integral: Tao: Chapter 1: sec.1.3. sec.1.5, Royden: Chapter 3: sec.3.6. Chapter 4

Lecture notes , Lecture notes , Problems (Due Feb. 19) solution

Week 7: Lebesgue integral continued: Tao: Chapter 1: sec.1.3. sec.1.4, Royden: Chapter 4

Problems (Due Feb. 26) solution

Week 8: Lebesgue integral continued: convergence theorems, approximation by continuous functions, Fubini and Tonelli Theorems. Tao: Chapter 1: sec.1.3, 1.4, 1.7. Royden: Chapter 4. Chapter 12. sec. 4

Problems (Due March 12) solution

Week 9: Fubini Theorem, Differentiation of monotone functions, BV functions Royden: Chapter 5. sec. 1, sec. 2.

Problems (Due March 19) solution

Week 10: BV functions, Absolution continuity. Royden: Chapter 5. Homework: Exercises 1.6.48 of Tao: an introduction to measure theory, Additional Problems (Due March 26) , solution

Week 11: L^p Spaces. Royden: Chapter 6, Tao: An Epsilon of Room, I, Chapter 1. sec. 1.3.

Problems (Due Apr. 2) , solution

Week 12: L^p Spaces, convolution, Maximal functions. Royden: Chap. 6, Tao: An Epsilon of Room, I, Chapter 1. sec. 1.3. Tao: An introduction to measure theory, Chapter 1, sec. 1.6. Problems (Due Apr. 9) , solution

Week 13: Read Royden Chap 11 sec. 1-3, sec. 7. Measure theory. Royden: Chap. 11. sec. 1, Chap 12. Tao: An introduction to measure theory, Chapter 1, sec. 1.4, sec. 1.7. Problems (Due Apr. 16)

Week 14: Caratheodory extension theorem, Lebesgue-Stieltjes and Riemann-Stieltjes integrals, product measure, dual of L^p. Royden: Chap 12. sec. 1-4, Chap 6. sec. 5 Tao: An introduction to measure theory, Chapter 1, sec. 1.7. Problems

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