MATH 6337 (Real Analysis I), Fall 2007

- Errata list for our textbook
- General Reading List
- Short review of Lebesgue measure
- Short review of Banach and Hilbert spaces
- Book: C. Heil, Five Mini-Courses on Analysis, Birkhäuser, Boston, in preparation.

- Chapter 0
- Chapter 1:
- Section 1.1: Introduction
- Section 1.2: Sigma-algebras
- Section 1.3: Measures
- Section 1.4: Outer measures (revised 9/18/07)
- Section 1.5: Borel measures on the real line

- Chapter 2:
- Section 2.1: Measurable functions
- Section 2.2: Integration of nonnegative functions
- Section 2.3: Integration of complex functions
- Section 2.4: Modes of convergence,
part 1 and
part 2
- Short review of metrics, norms, and convergence

- Section 2.5: Product measures
- Section 2.6: The n-dimensional Lebesgue integral

- Chapter 3:
- Section 3.1: Signed measures
- Section 3.2: The Lebesgue-Radon-Nikodym Theorem
- Section 3.3: Complex measures
- Section 3.4: Differentiation on Euclidean space
- Section 3.5, part a: Functions of bounded variation
- Section 3.5, part b: Singular and absolutely continuous functions
- Section 3.5, part c: Convex functions and Jensen's inequality (not included on final exam)

- Homework 1 (revised and with new due date). Due: September 4, 2007.
- Homework 2. Due: September 11, 2007.
- Homework 3. Due: September 18, 2007.
- Homework 4. Due: October 4, 2007.
- Homework 5. Due: October 18, 2007.
- Homework 6. Due: October 25, 2007.
- Homework 7. Due: November 13, 2007.
- Homework 8. Due: December 4, 2007.

- Exam 1 is scheduled for September 25, 2007 (in class). The exam will cover Sections 1.1-1.5 in Folland's text, and the corresponding posted lecture notes. You are allowed to bring one 8.5x11 sheet of notes to the exam (you can write on both sides).
- Exam 2 has been rescheduled for November 1, 2007 (in class). The exam will cover Sections 2.1-2.5 in Folland's text, and the corresponding posted lecture notes. You are allowed to bring one 8.5x11 sheet of notes to the exam (you can write on both sides).
- Our Final Exam is scheduled for 8:00-10:50, December 12, 2007 (in our usual classroom). The exam is comprehensive, covering Sections 1.1-1.5, 2.1-2.5, 3.1-3.5 in Folland's text, and the corresponding posted lecture notes. You are allowed to bring one 8.5x11 sheet of notes to the exam (you can write on both sides).

- 1.2 #2, 3 (part a is hard), 4
- 1.3 #6, 7, 8, 9, 10, 11, 12, 13, 14, 15
- 1.4 #17, 18, 19, 20, 23, 24
- 1.5 #25, 26, 28, 29, 31
- 2.1 #2, 3, 4, 5, 6, 8, 9, 10
- 2.2 #12, 13, 14, 15, 16, 17
- 2.3 #18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30
- 2.4 #32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 44
- 2.5 #46, 48, 50, 51
- 2.6 #55, 56, 57, 58, 59
- 3.1 #1, 2, 3, 4, 5, 6, 7
- 3.2 #8, 9, 10, 11, 12, 13, 16
- 3.3 #18, 19, 20, 21
- 3.4 #22, 23, 24, 25
- 3.5 #27, 31, 33, 35, 37

- Chapter 2 (Functions of bounded variation): #1, 2, 3, 4, 5, 6, 7, 8
- Chapter 3 (Lebesgue measure): #4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 23
- Chapter 4 (Measurable functions): #1, 2, 4, 5, 6, 12, 15, 16, 17
- Chapter 5 (The Lebesgue integral): #1, 2, 3, 4, 5, 7, 9, 10, 11, 12, 13, 20, 21
- Chapter 6 (Repeated integration): #1, 2, 3, 4, 6, 11
- Chapter 7 (Differentiation): #2, 6, 7, 8, 9, 10, 11