MAT 1000 / MAT 457 University of Toronto 201112
MAT 1000 / MAT 457 Real Analysis I, Fall 2012
Almut Burchard, Instructor
How to reach me: Almut Burchard, 215 Huron, # 1024, 64174.
 almut @math ,
www.math.utoronto.ca/almut/
 Lectures MWF 12:101PM, BA 6183 .
 Office hours Monday 34:30 and 5:356:30pm
Teaching assistant: Daniel Soukup,
daniel.soukup @mail.utoronto.ca .
Textbook:
G. Folland, Real Analysis: Modern Techniques and their Applications.
Wiley (either edition)

Measure Theory: abstract measures and σalgebras,
Monotone Class Theorem, outer measures and Caratheorodry's theorem,
Borel sets and Lebesgue measure
 Integration: convergence theorems, Fubini's theorem,
change of variables and polar coordinates in
R^{n}
 Lebesgue Differentiation:
HardyLittlewood maximal function,
density points, RadonNikodym theorem
 Functional Analysis: Hilbert spaces, orthonormal bases,
Riesz representation theorem, compact operators,
L^{p} spaces
Hölder and Minkowski inequalities
We will also consult other sources, including
 Eliott H. Lieb and Michael Loss, Analysis.
AMS Graduate Texts in Mathematics, Vol 14 (either edition)
 Elias Stein and Rami Shakarchi, Real Analysis.
Princeton Lectures in Analysis, Vol. 3
 A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis.
Dover Publications
 H.L. Royden: Real Analysis, Macmillan, 1988.
All of these are on reserve in the Mathematics library.
I will post additional notes on the web, as needed.
Evaluation:
 40% : Homework: weekly exercises
(due Wednesdays in class), plus a 5page essay
(due Wednesday December 19)
 20% : midterm test
(Wednesday November 7, 57pm; 2202 Sanford Fleming)
 40% : final examination (Wednesday,
December 12, 25pm 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.
Schedule:
Week 1 (September 1014)
Chapter 1  Measures
(Lieb & Loss, Chapter 1)
 M: Why do we need integration and measure theory?
 W: σalgebras
 F: The Monotone Class Theorem
Assignment 1
(due September 19)
Comments and hints
Week 2 (September 1721)
Chapter 1  Measures
 M: The Borel σalgebra. 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 2428)
Chapter 2  Integration
 M: Proof of Carathéodory's theorem
 W: Some corrections. Premeasures and elementary families
 F: Lebesgue measure on
R^{n}:
construction and geometric properties
Assignment 3
(due October 3)
Week 4 (October 15)
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)
Amendments
Week 5 (October 812)
Chapter 2  Integration
 M: Thanksgiving holiday
 W: Fatou's Lemma and Dominated Convergence
 F: The space L^{1}
Assignment 5
(due October 17)
Corrections and comments.
Week 6 (October 1519)
Chapter 6  L^{p}spaces
(Lieb & Loss, Chapter 2)
 M:
Norms and unit balls.
L^{p} spaces
 W:
L^{p} (R^{n}) for finite p: Smooth functions are dense,
translation is continuous
 F: Hölder's inequality;
the simplest interpolation inequality
Assignment 6
(due October 24)
Corrections
Week 7 (October 2226)
Chapter 2  Integration
 M: Product measures and Fubini's theorem
 W: Fubini's theorem, cont'd.
 F: Change of variables in
R^{n}
Assignment 7
(due October 31)
Week 8 (October 29November 2)
Chapter 2  Applications of measure and integration theory
 M: Integration in polar cordinates
 W: Volume and surface area of the
unit ball in R^{n}
 F: Littlewood's Three Principles
(no assignment due on November 7)
Week 9 (November 59)
... nothing new ...
 M: Infinite products
 W: Midterm test
57pm (2202 Sanford Fleming)
 F: No lecture
Choose essay topic (oneparagraph outline due on November 14)
Last year's midterm
test
Week 10 (November 1216)
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)
Corrections
Week 11 (November 1923)
Chapter 3  Signed measures and differentiation
 M: Signed measures and complex measures
 W: The LebesgueRadonNikodym theorem
 F: Differentiation in Euclidean space. The
HardyLittlewood maximal function
Assignment 9
(due November 28)
Corrections and clarifications
Week 12 (November 2630)
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)
Correction
Week 13 (December 26)
Chapter 3  Signed measures and differentiation
 M: The Fundamental Theorem of Calculus
 W: Fundamental Theorem of Calculus, cont'd
Wednesday December 12
 Final Exam 25pm (WI 1017)
(last year's exam,
UVa problems)
Essays (due December 19, my box)
Office hours December 17/18:
Mon 125pm, Tue 35pm.
The University of Toronto is committed to accessibility. If you
require accommodations for a disability, or have any accessibility
concerns about the course, the classroom or
course materials, please contact Accessibility Services as
soon as possible: disability.services@utoronto.ca, or
http://studentlife.utoronto.ca/accessibility