MATH 245A : Real Analysis

Course description: Measure theory (including construction of Lebesgue measure, and abstract measure theory), integration theory (convergence theorems, differentiation theorems, Fubini-Tonelli theorem). Littlewood's three principles. Possible other topics as time permits.

Announcements:

Final grades and distribution are available on my.ucla.edu. The final exams are being kept by Jim Ralston (I will be away until mid-March), but can be collected on request.

Instructor: Terence Tao, tao@math.ucla.edu, x64844, MS 6183
Lecture: MWF 2-2:50, MS 6229
Quiz section: Tu 2-2:50, MS 6229
Office Hours: Th 2-2:50, MS 6183
TA: Sungjin Kim, i707107@math.ucla.edu, MS 6160
TA Office hours: Tu 1-2, 3-4
Textbook: "Real Analysis: measure theory, integration, and Hilbert spaces", by Stein and Shakarchi. I will also post lecture notes on my blog site. Folland's "Real analysis" may also be used as an alternate text, but it is not required.
Prerequisites: Math 121, 131A, 131B (or equivalent). In particular, students should be familiar with the basic theory of the Riemann integral and concepts such as pointwise and uniform convergence of functions, of conditional and absolute convergence of series, and with the notions of open and closed sets on R^n (and ideally also on metric spaces and topological spaces). For instance, one should be able to explain why the sum of the functions sin(nx)/n^2, as n ranges from 1 to infinity, is uniformly convergent. Some minimal level of set theory - enough to understand phrases such as .a countable union of open sets. - would also be highly desirable. We will however briefly review this material whenever it is needed in class.
Grading: Homework (30%), Midterm (30%), Final (40%). In addition, a nominal bonus point (1%) will be awarded to each student who presents at least one homework problem at the blackboard during at least one quiz section..
Reading Assignment: We will cover most of Chapters 1,2,3,6 of Stein-Shakarchi (not necessarily in that order); see also Chapters 1-3 of Folland for a slightly different take on the same material. Lecture notes will be also provided on my blog site. Students are encouraged to comment on these posts. Note that the notes will cover more material than the lectures. Students are strongly encouraged to read some combination of the text or the lecture notes, independently of the homework assignments, either contemporaneously with or slightly ahead of the treatment of the material in lectures.
Homework: Homework will be assigned weekly from a combination of the text and the lecture notes. Homework may be turned in either in paper format, or electronically by email. When answering an exercise from the notes, you may use any results from previous exercises, previous notes, and the text. You can also use results from external sources (e.g. Folland) as long as they are properly cited. Three questions from each homework assignment will be graded in detail.

Homework 1 (due Friday, Oct 8):

245A prologue, Exercise 3 (uniqueness of elementary measure).
245A prologue, Exercise 8.1 (every triangle is Jordan measurable).
245A prologue, Exercise 25 (area interpretation of the Riemann integral), assuming that f is non-negative. (You are of course welcome to attempt the general case, when f is signed, if you wish.)
245A Notes 1, Exercise 10 (Cantor set).
Stein-Shakarchi, Chapter 1, Page 39, Exercise 5(ab).

Homework 2 (due Friday, Oct 15):

245A Notes 1, Exercise 17 (Carathedory criterion for Lebesgue measurability).
245A Notes 1, Exercise 25 (Non-additivity of Lebesgue outer measure).
245A Notes 2, Exercise 12 (Upper Lebesgue integral and outer Lebesgue measure).
245A Notes 2, Exercise 19 (Linearity of the absolutely convergent integral).
Stein-Shakarchi, Chapter 1, Page 44, Exercise 28 (Dense intervals).
Stein-Shakarchi, Chapter 1, Page 44, Exercise 29 (Steinhaus theorem).

Homework 3 (due Friday, Oct 22):

245A Notes 2, Exercise 6 (Measurability and the region under a graph).
245A Notes 2, Exercise 13 (Area interpretation of Lebesgue integral).
245A Notes 2, Exercise 24 (Measurability and pointwise limits of continuous functions).
Stein-Shakarchi, Chapter 2, Page 91, Exercise 10 (Integrability of power laws via the horizontal wedding cake decomposition). Errata: in this question, you should assume that f is measurable.

Homework 4 (due Friday, Oct 29):

245A Notes 3, Exercise 4 (Classification of finite Boolean algebras)
245A Notes 3, Exercise 17 (Product of Borel sets is Borel)
245A Notes 3, Exercise 18 (Slices of Borel sets are Borel)
245A Notes 3, Exercise 23 (Subadditivity and convergence for countably additive measures)

Homework 5 (due Friday, Nov 5):

245A Notes 3, Exercise 33 (Inclusion-exclusion principle)
245A Notes 3, Exercise 36 (Change of variables formula)
245A Notes 3, Exercise 37 (Pushforward by linear transformations)
Stein-Shakarchi, Chapter 6, Page 312, Exercise 2 (Completion of a measure space). Errata: the formula for E_1^c in the hint is not quite correct.

Homework 6 (due Friday, Nov 12):

245A Notes 3, Exercise 45 (Almost dominated convergence)
245A Notes 3, Exercise 46 (Defect form of Fatou's lemma)
245A Notes 4, Exercise 11 (Uniform integrability and integrals on small sets)

Homework 7 (due Friday, Nov 19):

245A Notes 4, Exercise 16 (Pointwise almost everywhere convergence is equivalent to almost uniform convergence assuming domination). Note: this exercise may not be present or incorrect in versions of these notes prior to Nov 11.
245A Notes 5, Exercise 8 (Steinhaus theorem)
245A Notes 5, Exercise 9 (Measurable homomorphisms). The third part of the exercise is optional, as we have only covered Zorn's lemma in passing.
Stein-Shakarchi, Chapter 3, Page 147, Exercise 12 (Differentiable function with non-integrable derivative)

Homework 8 (due Tuesday, Nov 30 - note change of date):

245A Notes 5, Exercise 26 (Approximations to the identity). Note: there are typos in this exercise for any version of the notes dated Nov 22 or earlier. Also, part 5 of the question relies on Tonelli's theorem, which has not yet been covered, so I am removing it from the assignment.
245A Notes 5, Exercise 46 (Cantor function). Note: there are typos to part (6) for any version of the notes dated Nov 24 or earlier.
Stein-Shakarchi, Chapter 3, Page 146, Exercise 5 (Integrable function with non-locally-integrable maximal function)

There will be no graded homework assignments beyond Homework 8. However, I provide below a list of questions that I would recommend that you attempt as preparation for the final; at least one of these questions will actually appear on the final itself (possibly in a slightly modified form).

245A Notes 5, Exercise 41 (Second derivative of convex functions)
245A Notes 5, Exercise 47 (Absolute continuity in relation to other regularity concepts)
245A Notes 6, Exercise 6 (Not all finitely additive measures are premeasures)
245A Notes 6, Exercise 9 (Approximation by elementary sets)
245A Notes 6, Exercise 20 (Product of Dirac and counting measures)
245A Notes 6, Exercise 23 (Counterexample to Fubini's theorem in the non-absolutely-integrable case)
245A Notes 6, Exercise 25 (Distribution formula)
Stein-Shakarchi, Chapter 3, Page 150, Exercise 25 (Non-negative Dini derivative implies monotonicity)
Stein-Shakarchi, Chapter 6, Page 314, Exercise 14 (Construction of product measure for more than two factors)

Exams: The midterm will be in class on Wednesday, Oct 27, and consist of three questions, which will be drawn from either the blog exercises or from Stein-Shakarchi; this is open book, open notes (but no electronic assistance is allowed). In particular, doing the exercises from these sources will be an excellent preparation for the midterm. The final is on Friday, Dec 10, from 8am-11am, and will consist of approximately seven questions, about half of which will be drawn from the blog exercises or from Stein-Shakarchi. This is also open book and open notes.

Online resources:

The blog for this course.