MAT 1000 / MAT 457 University of Toronto 2011-12

MAT 1000 / MAT 457 Real Analysis I, Fall 2012

Almut Burchard, Instructor

How to reach me: Almut Burchard, 215 Huron, # 1024, 6-4174.
almut @math ,
Lectures MWF 12:10-1PM, BA 6183 .
Office hours Monday 3-4:30 and 5:35-6:30pm
Teaching assistant: Daniel Soukup, daniel.soukup .
Textbook:   G. Folland, Real Analysis: Modern Techniques and their Applications. Wiley (either edition) We will also consult other sources, including All of these are on reserve in the Mathematics library. I will post additional notes on the web, as needed.
40% : Homework: weekly exercises (due Wednesdays in class), plus a 5-page essay (due Wednesday December 19)
20% : midterm test (Wednesday November 7, 5-7pm; 2202 Sanford Fleming)
40% : final examination (Wednesday, December 12, 2-5pm in WI 1017)
Remarks. Please discuss lectures and homework problems among yourselves and with me, and consult other sources. But write up your assignments in your own words, and be ready to defend them! Your work will be judged on the clarity of your presentation as well as correctness and completeness.


Week 1 (September 10-14)
Chapter 1 -- Measures (Lieb & Loss, Chapter 1)
M: Why do we need integration and measure theory?
W: σ-algebras
F: The Monotone Class Theorem
Assignment 1 (due September 19)
Comments and hints
Week 2 (September 17-21)
Chapter 1 -- Measures
M: The Borel σ-algebra. Measures
W: Measures and outer measures
F: Carathéodory's theorem
Assignment 2 (due September 26)
Comments and hints, Solution to Problem 2
Week 3 (September 24-28)
Chapter 2 -- Integration
M: Proof of Carathéodory's theorem
W: Some corrections. Premeasures and elementary families
F: Lebesgue measure on Rn: construction and geometric properties
Assignment 3 (due October 3)
Week 4 (October 1-5)
Chapter 2 -- Integration
M: Measurable functions
W: Construction of the integral. Monotone Convergence
F: Simple functions and really simple functions
Assignment 4 (due October 10)
Week 5 (October 8-12)
Chapter 2 -- Integration
M: Thanksgiving holiday
W: Fatou's Lemma and Dominated Convergence
F: The space L1
Assignment 5 (due October 17)
Corrections and comments.
Week 6 (October 15-19)
Chapter 6 -- Lp-spaces (Lieb & Loss, Chapter 2)
M: Norms and unit balls. Lp -spaces
W: Lp (Rn) for finite p: Smooth functions are dense, translation is continuous
F: Hölder's inequality; the simplest interpolation inequality
Assignment 6 (due October 24)
Week 7 (October 22-26)
Chapter 2 -- Integration
M: Product measures and Fubini's theorem
W: Fubini's theorem, cont'd.
F: Change of variables in Rn
Assignment 7 (due October 31)
Week 8 (October 29-November 2)
Chapter 2 -- Applications of measure and integration theory
M: Integration in polar cordinates
W: Volume and surface area of the unit ball in Rn
F: Littlewood's Three Principles
(no assignment due on November 7)
Week 9 (November 5-9)
... nothing new ...
M: Infinite products
W: Midterm test 5-7pm (2202 Sanford Fleming)
F: No lecture
Choose essay topic (one-paragraph outline due on November 14)
Last year's midterm test
Week 10 (November 12-16)
Chapter 5 -- Functional Analysis
M: Fall Break
W: The Kolmogorov Extension Theorem (correctly, this time)
F: Excursion: The Baire Category Theorem (lecture by Daniel Soukup)
Assignment 8 (due November 21)
Week 11 (November 19-23)
Chapter 3 -- Signed measures and differentiation
M: Signed measures and complex measures
W: The Lebesgue-Radon-Nikodym theorem
F: Differentiation in Euclidean space. The Hardy-Littlewood maximal function
Assignment 9 (due November 28)
Corrections and clarifications
Week 12 (November 26-30)
Chapter 3 -- Signed measures and differentiation
M: Lebesgue density
W: Excursion: Applications of the Baire Category Theorem (lecture by Daniel Soukup)
F: Functions of bounded variation
Assignment 10 (due Friday December 7, in Daniel's mailbox)
Week 13 (December 2-6)
Chapter 3 -- Signed measures and differentiation
M: The Fundamental Theorem of Calculus
W: Fundamental Theorem of Calculus, cont'd
Wednesday December 12
Final Exam 2-5pm (WI 1017)
(last year's exam, UVa problems)
Essays (due December 19, my box)
Office hours December 17/18: Mon 12-5pm, Tue 3-5pm.

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