Math 533  Real Analysis, Fall 2010

General Information

• Folland "Real Analysis" 2nd edition (main text). [Ch 0, Ch 1]
• Rudin "Real and Complex Analysis" (optional).

Syllabus (tentative)

Measures (5 weeks): sigma-algebras, measures, outer measure, Caratheodory extension theorem, Lebesgue and Lebesgue-Stieltjes measures on the line.

Integration (3.5 weeks): measurable functions, simple functions, L⁺ and L¹, Lebesgue Monotone and Dominated convergence theorems, modes of convergence, theorems of Egoroff and Lusin, product measures and Tonelli-Fubini theorems.

Signed measures (1.5 weeks): Hahn decomposition, Jordan decomposition, Radon-Nikodym theorem.

Intro to Functional Analysis (2.5 weeks): Normed spaces, linear functionals, L^p spaces (Holder, Minkowsky, dual space), Riesz representation theorem, weak convergence, intro to Hilbert spaces.

Date	Topic	Homework	Comments
8/23 8/25 8/27	Introduction Elementary set theory algebras/sigma-algebras	Read Chapter 0, especially 0.1, 0.3, 0.5 HW1 (due W 9/1) .	. Text Ch 0 .
8/30 9/1 9/3	algebras/sigma-algebras measures measure spaces	Section 1.2; Problems 1-4 on p.24 Section 1.3; Problems 8-12, 13-16 (opt) Section 1.3-4	Text Ch 1 HW1 is due .
9/6 9/8 9/10	Labor Day completeness outer measures	Rest Section 1.3; Problems 14, 15 .	Rest . HW2 is due: 3, 8, 10, 12
9/13 9/15 9/17	Caratheodory Lebesgue Lebesgue-Stieltjes	Section 1.4; Problems 17-19, 22-24 (p.32) Section 1.5 Section 1.5	. . HW3 is due: 14, 18, 19, 24.
9/20 9/22 9/24	Lebesgue-Stieltjes Regularity properties Cantor-Lebesgue function	Monotonic functions, Lebesgue-Stieltjes . .	A monotonic function with prescribed countable disc. set . HW4 is due: 28, 30, 31, 32.
9/27 9/29 10/1	Measurable functions Integration of simple functions Integration of positive functions	Section 2.1 Problems: 1-10 (p.48/9) Section 2.1 .	HW4 is due: 28, 30, 31, 32. HW5 is due: 2, 7, 8
10/4 10/6 10/8	Monotone Convergence Theorem L^1 functions Properties of the integral	Section 2.2 Section 2.2; Problems: 13-17 Section 2.3	HW5 is due: 2, 7, 8, 31(p.40) using regularity . .
10/11 10/13 10/15	Fatou's Lemma and DCT Approximations Lebesgue vs. Riemann	. .	. . HW6 is due: 13, 14, 16, 18.
10/18 10/20 10/22	Integration Review Midterm, part I Modes of convergence	. . Section 2.4	. HW7 is due: 20, 21, 24, 28. MT, part II take home
10/25 10/27 10/29	Midterm (part I) Cauchy in measure Completeness of L^1, Egoroff	rescheduled Read 2.4, Problems: 32-34, 36, 38 .	. Midterm (part II) is due .
11/1 11/3 11/5	Product measures Tonelli-Fubini The n-dim Lebesgue integral	Section 2.5 Section 2.5 (cont.), Problems: 46, 48, 50, 51 Read section 2.6; Problems: 59, 60	HW7 is due: 33, 34, 38, 41 HW8 is due: 33, 34, 38, 41 .
11/8 11/10 11/12	Signed measures, Hahn decomposition Jordan decomposition Lebesgue Radon-Nikodym thm	Section 3.1 Section 3.1; Problems: 1-5, 7 Problems: 9, 12, 13, 16, 17. Jensen inequality: Problem 42 p.109.
11/15 11/17 11/19	Absolute continuity L^p spaces, Holder and Minskowski Completeness	absolutely continuous functions and further remarks Section 6.1, Problems: 5, 6, 9, 10 (p.187) .	HW9 is due: 3, 4, 16, 17 HW9 is due: 3, 4, 16, 17 .
11/22 11/24 11/26	Interpolation, approximations The dual of L^p Thanksgiving	C_c(R) is dense in L^p Section 6.2; Problems: 18, 20, 22 No classes	. . No classes
11/29 12/1 12/3	Riesz representation Weak convergence, Hilbert spaces Hilbert spaces	Signed measures as functionals on C(X) (see Sec 7.3) Section 5.5 Section 5.5 (cont.), Conditional expectation	Notes on Riesz Solutions to midterm Part I, Part II HW10 is due: 9, 10 (p.187), 18, 22 (p.192)
12/9	Final Exam, Thursday 10:30-12:30	Good luck

Grading

Announcements

• Midterm is on Friday, October 22.
• Final exam is on Thursday, December 9, 10:30-12:30.
• Terence Tao: useful tricks in analysis
• Solutions to midterm Part I, Part II