MATH 245A : Real Analysis


        (Dec 17) Final solutions are available here. Final grades are available here.

        (Dec 8) For finals week, my office hours will be Wed 9-12, Alex’s office hours are Tu-Th 1-3.

        (Nov 23) Sixth Homework postponed to Nov 30, since we are approximately one lecture behind.

        (Nov 3) The midterm results have been cancelled. What this means is that the grading of the course will now consist of 30% homework, 70% final instead of 30% homework, 30% midterm, 40% final. This is due to serious problems with the midterm which have rendered it impossible to grade it in a fair manner; in particular, Question 1 is false as stated and those students who were being careful to be rigorous everywhere would have been heavily penalized for wasting too much of their time on this question. Also Q2(b) and Q3 are more difficult than anticipated. Some partial solutions and discussion to the midterm can be found here.

        (Nov 1) Fourth Homework postponed to Nov 9, since we are approximately one lecture behind.

        (Oct 25) TA office hours corrected.

        (Oct 2) The date and place of the final has been corrected.

        (Sep 30) The class cap has been raised from 50 to 55. This is the last and final class size change; once this cap has been reached, the class will be closed. 55 is the seating capacity of the new room and we will not be admitting any further enrolments.

        (Sep 29) The class cap has been raised from 39 to 50, and is now re-open. Also, the classroom has moved, to Hershey 1651 (TA session in Boelter 9436).

        (Sep 17) The class cap has now been raised from 35 to 39, and is now closed. The first lecture is on Oct 1.

        (Sep 14) The class cap has now been raised from 30 to 35. If you still need a PTE or have other problems enrolling, contact me.


        Instructor: Terence Tao,, x64844, MS 5622

        Lecture: MWF 12-12:50, Hershey 1651. Note change of room.

        Quiz section: Tu 3-3:50, Boelter 9436. Note change of room.

        Office Hours: Mon 2-3, Thu 11-12.

        TA: Alex Smith,, MS2963

        TA Office hours: Tu 9-10, 12-1; Thu 12-1, or by appointment

        Textbook: Folland, Real Analysis, Second Edition, Wiley Interscience 1999, ISBN 0471317160. We will cover Chapters 1-3 (Measure, integration, and differentiation theory); some variation from this plan may develop depending on time constraints. You should read Chapter 0 (set theory and the real number system) yourself; we will use it as necessary in the course and will assume you have some familiarity with this material. Note that a current list of errata to this text is maintained here (thanks to Julia Garibaldi for pointing this out).

        Prerequisite: 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). 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 (and you should also review Chapter 0 of the textbook and be comfortable with that material).

        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.

        Exams: The midterm will be on Wednesday, Nov 3; it will be three questions, covering Chapter 1 and Chapter 2.1. The exam is open book and open notes. The final will be on Friday Dec 17, at 8:00 am, Hershey 1651, with nine questions covering all topics up to and including Chapter 3.3 (Lebesgue-Radon-Nikodym theorem). Only the best seven of nine questions will be counted for the final grade. The final is open book and open notes. Makeup exams will only be available by prior arrangement or by an exceptionally good excuse after the fact. Requests to reweight the exam gradings also require an exceptionally good excuse (i.e. better than “I did poorly on my midterm”).

        Reading Assignment: You should read Sections 0.1, 0.3, 0.4, 0.5; this should be material that you have seen in other courses, but we will need it quite a bit in our lectures. Regarding the material on the axiom of choice (Section 0.2), we will review it separately in class if and when it comes necessary. Of course, you should also be reading the sections covered in lecture (and homework) concurrently with the course.

        Homework: There will be seven homework assignments, due in Quiz sections. In each assignment, three of the homework questions, selected at random, will be graded in detail. NOTE: In your assignments you may freely use the axiom of choice.

1.      First homework (Due Tuesday, October 12): Folland Chapter 1, Questions 3, 4, 5, 7, 8, 12, 14, 15

1.      Errata: In Q3a, “disjoint sets” should be “disjoint non-empty sets”. Hint: If the algebra contains an infinite nested sequence of strictly decreasing sets, then you are done, so you can assume that such sequences do not exist. Use this to construct an “atom” – a non-empty measurable set which has no measurable proper subset. Now find a lot of such atoms.

2.      Second homework (Due Tuesday, October 19): Folland Chapter 1, Questions 13, 17, 19, 25, 26, 27, 29

3.      Third homework (Due Tuesday, October 26): Folland Chapter 1, Questions 18, 30, 31, 33; Folland Chapter 2, Questions 2, 3, 4, 8

4.      Fourth homework (Due Tuesday, November 9) - note change of date: Folland Chapter 2, Questions 9, 10, 13, 14, 15, 16, 19, 20

5.      Fifth homework (Due Tuesday, November 16): Folland Chapter 2, Questions 21, 23, 25, 26, 34, 36, 39, 42

6.      Sixth homework (Due Tuesday, November 30) - note change of date: Folland Chapter 2, Questions 44, 46, 49, 50, 54, 56, 59

7.      Seventh homework (Due Tuesday, December 7): Folland Chapter 3, Questions 1, 4, 5, 9, 13, 16, 17

1.      Errata: In Q16, mu and nu need to be assumed to be sigma-finite.