MAT 1000 / MAT 457 University of Toronto Fall 2013
 

MAT 1000 / MAT 457 Real Analysis I, Fall 2013

Almut Burchard, Instructor

How to reach me: Almut Burchard, 215 Huron, # 1024, 6-4174.
almut@math, www.math.utoronto.ca/almut/
Lectures MWF 12:10-1PM, BA 6183 .
Office hours Mondays 4:15-6:30
Teaching assistant: Daniel Soukup, daniel.soukup @mail.utoronto.ca .
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, as needed.
Evaluation:
40% : Homework: weekly exercises (due Wednesdays in class), plus a 5-page essay (due Wednesday November 6)
20% : midterm test (Wednesday October 30, evening)
40% : final examination (Wednesday, December 11, afternoon)
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.

Schedule:

Week 1 (September 9-13)
Chapter 1 -- σ-algebras (L&L, Chap. 1; S&S, Introduction)
M: Why do we need integration and measure theory?
W: Additive and σ-additive set functions. Banach-Tarski paradox
F: The Monotone Class Theorem
Assignment 1 (due September 18, in class)
Corrections and comments
Week 2 (September 16-20)
Chapter 1 -- Measures
M: The Borel σ-algebra
W: Outer measures
F: Carathéodory's extension theorem
Assignment 2 (due September 25, in class)
Corrections and comments
Week 3 (September 23-27)
Chapter 1 -- Lebesgue measure
M: Construction of Lebesgue measure on Rn
W: Borel measures on the real line
F: The Vitali-Hahn-Saks theorem (Notes)
Assignment 3 (due October 2, in class)
Comments
Week 4 (September 30 - October 4)
Chapter 2 -- Integration: The great convergence theorems
M: Measurable functions. Construction of the integral
W: Monotone Convergence and Fatou's lemma
F: Dominated Convergence. Examples
Assignment 4 (due October 9, in class)
Week 5 (October 7-11)
Chapter 2 -- Integrable functions
M: Simple functions and really simple functions
W: Linearity and translation invariance of the integral
F: The space L1
Assignment 5 (due October 16, in class)
Comments
Week 6 (October 14-18)
Chapter 2 -- Fubini's theorem
M: Thanksgiving holiday
W: Product measures and Fubini's theorem
F: Change of Variables
Assignment 6 (due October 23, in class)
Corrections
Week 7 (October 21-25)
Chapter 2 -- Applications in Rn
M: Proof of the Fubini-Tonelli theorem
W: Integration in polar cordinates. Gaussians and the Gamma function
F: Volume and surface area of the unit ball in Rn
(No assignment due October 30)
Week 8 (October 28-November 1)
Chapter 2 -- Applications
M: Infinite products and Kolmogorov's extension theorem
W: Midterm test (2011, 2012, practice problems)
F: Littlewood's Three Principles
Office hours this week: M 4:30-6:30, T 3-4, W 12-1
Essay (due November 6, in class or to Almut's mailbox)
Week 9 (November 4-8)
Excursion -- Fourier series (S&S, Section 4.3)
M: Periodic functions. The space L2(0, 2π)
W: Fourier coefficients
F: The Poisson kernel
Assignment 7 (due November 13, in class)
Corrections
Week 10 (November 11-15)
Chapter 3 -- Signed measures
M: Fall break (no lecture)
W: Major correction: convergence of Fourier series
F: Signed measures and complex measures
Assignment 8 (due November 20, in class)
Comments
Week 11 (November 18-22)
Chapter 3 -- Signed measures: Radon-Nikodym Theorem
M: Hahn and Jordan decompositions of a signed measure
W: Mutually singular and absolutely continuous measures
F: The Lebesgue-Radon-Nikodym theorem
Assignment 9 (due November 27, in class)
Comments
Week 12 (November 26-30)
Chapter 3 -- Differentiation in Euclidean space
M: The Hardy-Littlewood maximal function
M: The Lebesgue Differentiation Theorem
F: Lebesgue density
Assignment 10 (due Friday December 6, noon in Daniel's mailbox)
Week 13 (December 2-6)
Chapter 3 -- The Fundamental Theorem of Calculus
M: BV functions and functions and AC functions
Wednesday December 11: Final Exam (2-5pm. BA 6183)
Old exams: (2011, 2012, additional problems)

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