MAT 1000 / MAT 457 Real Analysis I, 2011-12

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 Monday afternoons (between 2 and 6, except during seminars)

Teaching assistant: Mustazee Rahman mustazee.rahman @ utoronto.ca .

Textbook:   Elias Stein and Rami Shakarchi, Measure Theory, Integration, and Hilbert Spaces

• Measure Theory: Lebesque measure and integration, convergence theorems, Fubini's theorem, Lebesgue differentiation theorem, abstract measures, Caratheodory theorem, Radon-Nikodym 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, 14 (either edition),
• G. Folland, Real Analysis: Modern Techniques and their Applications, Wiley (either edition),
• A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis, 1975,
• 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, by 4pm sharp in Mustazee's mailbox), plus 2 challenge problems or an essay of your choice (ca. 5 pages, can be handwritten or typed, anytime during the semester)

20% : midterm test (evening)

40% : final examination

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. Exemplary solutions will be posted (and will receive 1pt extra credit per problem).

Schedule:

Week 1 (September 12-16)
Chapter 1 -- Construction of Lebesgue measure

M: Why do we need integration and measure theory?

W: Rectangles

F: Exterior measure

Assignment 1 (due September 21): SS 1.6 Exercises 2, 4, 5. [Solutions (Vitaly Kuznetsov)]
Read: SS 1.1-3 (R 3.1-2, KF 25).

Week 2 (September 19-23)
Chapter 6 -- Carathéodory's theorem; Chapter 7 -- Hausdorff measure and dimension; Bob Jerrard's Lecture notes

M: Properties of Lebesgue measure: Outer regularity, translation invariance

W: General measure theory: Construction of a measure from an outer measure

F: Hausdorff measure and Hausdorff dimension

Assignment 2 (due September 28): SS 1.6 Exercises 6, 10, 16. [Solutions (Jennifer Vaughan)]
Read: SS 6.1 and 7.1;   Additional reading (optional): LL 1.1-4, F 1.1-3, KF 26-27.

Week 3 (September 26-30)
SS Chapter 1 -- More about Lebesgue measure

M: Construction of a non-measurable set

W: The Brunn-Minkowski inequality

F: Proof of Brunn-Minkowski

Assignment 3 (due October 5): SS 1.6 Exercises 21, 27, 37. [Solutions (Nikita Nikolaev)]
Read: SS 1.4-5.

Week 4 (October 3-7)
SS Chapter 2 and LL Chapter 1 -- Integration

M: The Borel sigma-algebra. Measurable functions

W: Definition of the integral

F: Simple functions and really simple functions (step functions)

Assignment 4 (due October 12): SS 1.6 Exercises 18, 28, SS 2.5 Exercise 9. [Solutions (Daniel Soukup)]
Read: SS 1.4, SS 2.1 (LL 1.5).

Week 5 (October 10-14)
SS Chapter 2 and LL Chaper 1 -- The great convergence theorems

M: Thanksgiving holiday

W: Linearity of the integral. Monotone convergence

F: Fatou's lemma, dominated convergence; the space L^1

Assignment 5 (due October 19): SS 2.5 Exercises 12, 15 (combine with LL Exercise 1.12); LL Exercise 1.7.
Read: SS 2.1 and 2.2, LL 1.6-1.9.

Week 6 (October 17-21)
SS Chapter 2 and LL Chaper 1 -- Fubini's theorem

M: Completeness of L^1

M: Fubini-Tonelli theorem for Borel functions

F: Monotone class theorem

Assignment 6 (due October 26): SS 2.5 Exercises 18, 21 [in (a), assume that f and g are a.e. finite;
in (c), f and g should be integrable], and 24, adding
(c): Let A and B be measurabe sets of finite positive measure. Define their essential sum
A+ess B to be the set where the convolution of their characteristic functions is positive.
Show that A+ess B is an open subset of A+B. In particular, it is measurable.
Remark: We will later show that the Brunn-Minkowski inequality holds for A+ess B.
[Solutions (Mary He)]

Week 7 (October 24-28)
F Section 2.6 -- Change of Variables

M: Fubini's theorem for Lebesgue measure

W: Change of variables

F: No lecture

Week 8 (October 31-November 4)
F Section 2.7 -- Polar coordinates

M: The uniform surface measure on the sphere. Volume and surface area of balls in R^d

W: More spherical integrals. Littlewood's three principles

Midterm Test November 2, 5-7pm, HA 410 (Haultain building)

F: The L^p-norms. Hölder's inequality.

Challenge Problems / Essays : Please submit a one-paragraph outline by Wednesday, November 9, 4pm
(in class or to my mailbox).

Week 9 (November 7-11)
SS Chapter 4 -- Hilbert spaces

M: Fall break

W: Completeness of L^p

F: Hilbert spaces

Assignment 7 (due November 16)

Week 10 (November 14-18)
SS Chapter 4 -- Hilbert spaces

M: Orthonormal bases.

W: Fourier series. The Poisson kernel.

F: Closed subspaces and orthogonal projections.

Assignment 8 (due November 23)
Please add to Problem 3d the assumption that there exists a constant c<1 such that ||P_2 h|| is bounded by c ||h||
for all h in S_1 that are orthogonal to S. If you apply this to f_n-Pf, you should get the convergence easily.
Make a sketch of the case where S_1 and S_2 are lines in the plane.

Gratuitous remark. The constant c = cos α can be interpreted as the cosine of the angle between the subspaces, defined by

c = \sup { | < h_1, h_2 >| : h_1 unit vector in S_1 orthogonal to S_2, h_2 unit vector in S_2 orthogonal to S_1 }.

For example, if S_1 and S_2 are hyperplanes, α is the angle between the normals, and if S_1 and S_2 are lines,
it's the angle between the lines. By Schwarz' inequality, c cannot exceed 1. If S_1 and S_2 have finite dimension,
then c<1 (it's defined by maximizing a continuous function over a compact set), and sequences converegs exponentially.
In infinite dimensions, c=1 is possible. Then convergence can be slow. Two elegant proofs of von Neumann's theorem:

• Netanyun and Solmon, MAA Monthly 113(7): p.644-648, 2006 (see p. 645)
• Bauschke, Matouskova, and Reich, Nonl. Anal. 46: p.715-738, 2004 (see p. 721).

Week 11 (November 21-25)
SS Chapter 3 -- Differentiation and integration on R^d

M: The Riesz representation theorem

W: Lebesgue's differentiation theorem. Maximal functions.

F: Density points of a set, Lebesgue points of a function. Approximations of the identity

Assignment 9 (due November 30): SS 3.5 Exercises 4, 26 (or 24); and
LL Exercise 2.10: Let f be a real-valued function on the real line that is additive, i.e., f(x+y)=f(x) + f(y) for all x,y.
(a) If f is locally integrable, prove that it is linear, i.e., f(x) = α x for some constant α;
(b) Prove the same conclusion, assuming only that f is measurable.
(Hint: Consider g(x) = e^{if(x)}, and convolve with an approximate identity)
(c) Argue that there exist non-measurable additive functions that are not linear.
Read: SS Chapter 3.
For the second problem, we'll accept either 26 or 24 until next week (Dec. 7)

Week 12 (November 28-December 2)
SS Chapter 3 -- Differentiation and integration for functions of a single variable

M: Functions of bounded variation

W: Total variation. Jump functions.

F: Differentiation of BV functions

Week 13 (December 5-7)
SS Chapter 6.4 -- Signed measures

M: Absolutely continuous functions.

W: The Radon-Nikodym theorem

Challenge Problems / Essays due December 7