Math 533  Real Analysis, Fall 2008

General Information

• Folland "Real Analysis" 2nd edition (main text).
• 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 (2 weeks): Hahn decomposition, Jordan decomposition, Radon-Nikodym theorem.

Elementary theory of Hilbert spaces (2.5 weeks): orthogonal projections and compliments, bases, Stone-Weierstrass, Intro to Fourier series.

Date	Topic	Homework	Comments
8/25 8/27 8/29	Introduction Elementary set theory algebras/sigma-algebras	Read Chapter 0, especially 0.1, 0.3, 0.5 HW1 (due W 9/3) .
9/1 9/3 9/5	Labor Day sigma-algebras + measures Measure spaces	Read 1.2-3, HW2 (due next week 9/12) Read 1.2-1.3
9/8 9/10 9/12	Measures, completeness outer measures Caratheodory, premeasures	Read 1.3 Read 1.4 start 1.5	HW3: 12-14 (p.27), 17-19, 22, 24 (p.32-33) due 9/22
9/15 9/17 9/19	Premeasure Lebesgue Lebesgue-Stieltjes	.	.
9/22 9/24 9/26	Regularity Cantor-Lebesgue function Review	HW3 is due . .	.
9/29 10/1 10/3	Midterm 1 Simple functions Integration of Simple	Midterm Read 2.1 Read 2.1	. Problems 1-10 (p.48/9) .
10/6 10/8 10/10	Integration of L^+ Monotone Convergence Thm Fatou, L^1 functions	Read 2.2, Problems: 13, 14, 16 (p.52) Read 2.3, Problems: 17 (p.52), 18-22 Problems: 25-27 (p.59)	. Tutorial: Midterm review .
10/13 10/15 10/17	Dom. Conv. Thm Riemann vs. Lebesgue Modes of Convergence	Read 2.3, learn Thms 2.27, 2.28 Read 2.4 Problems: 32-36 (p.63) .	. Tutorial: 13, 14 .
10/20 10/22 10/24	Modes of Convergence Egoroff's theorem' Product measures	. Luzin's Thm (44 on p.64) Problems: 37-41 (p.63), 45-48 (p.69)?	. Tutorial: 26, ... Extra Credit 1
10/27 10/29 10/31	Tonelli-Fubini Theorem Lebesgue on R^n Signed Measures	Read 2.5. Problems: 49-51 (p.69). Problems: 55, 56 (p.77) Start reading 3.1	Tutorial: 25, 44, 24 Tutorial: 33, 34, 36 .
11/3 11/5 11/7	Hahn and Jordan Lebesgue, Radon-Nikodym Midterm 2 !	Problems: 2-5, 7 (p.88) Problems: 9, 12 Good luck !	Tutorial: 44, 38 Tutorial: 40, 50, 51 MT2-II (take home)
11/10 11/12 11/14	Around R-N L^p spaces L^\infty	Problems: 16, 17, 21 (p.92) Read 6.1 (p.181-) .	Tutorial: 3, 4 (p.88) Tutorial: 5, 7, 12. .
11/17 11/19 11/21	The dual of L^p Riesz representation thm Hilbert spaces	Read 6.2. Problems 18, 20-22 (p. 192) Read Lemma 7.15 (p. 221)	Tutorial: MT2-I Tutorial: MT2-II .
11/24 11/26 11/28	Hilbert spaces Fourier series Thanks Giving	Read 5.5 . Turkey (optional)	Tutorial: Review Tutorial: Review Enjoy

Grading

Homework (25%) + 2 x Midterms (2x25%=50%) + Final (25%)
Midterm 1 (25%) + Midterm 2 (15+20=35%) + Final (40%)
Extra Credit Assignment 1(+ up to 5%): Problems 27, 28, 30, 31(a-d)

Announcements

• Room changed to 208 BH starting Friday 8/29.
• No classes on Monday, Sept 2 - Labor Day.
• Midterm 1 on Monday, September 29.
• Midterm 2 on Friday, November 7.
• Tutorials (MW 11-12 in 305 TH) have been added and are strongly recommended.