Math 201a: Real Analysis — Fall 2026
Professor:
Katy Craig,
katy•craig at ucsb • edu
Lecture:
Tuesday and Thursday, 11am-12:15pm,
Phelps Hall, Room 1508
Office Hours:
Wednesdays 12:45-1:45pm and Fridays 10-11am (SH 6507)
Textbook:
Folland,
Real Analysis: Modern Techniques and Their Applications,
second edition
Other Recommended References:
- Lieb and Loss, Analysis, second edition
- Rudin, Real and Complex Analysis, third edition
- Bogachev, Measure Theory
Exams: There will be two midterms and one final exam. The examinations will be closed book and closed note. There will be no retaking or rescheduling exams under any circumstances.
- First Midterm: Tuesday, October 20th, 11am-12:15pm
- Second Midterm: Thursday, November 12th, 11am-12:15pm
- Final Exam: Wednesday, December 9th, 12-3pm
Homework:
- Homework will be due Sundays at 11:59pm.
- Assignments will be posted on this website and submitted via Gradescope
- Only problems marked with an asterisk (*) should be submitted for grading.
- At least one problem on each of the exams will be chosen from the non-asterisked homework problems.
- No late homework will be accepted.
- The lowest two homework grades will be dropped and will not count toward the final grade.
-
Regarding collaboration/Google/AI:
- The solutions to most homework problems can be found on the internet. The purpose of homework is to practice solving problems. Don’t miss out on that practice, or you will deprive yourself of key preparation for the exams.
- Discussing homework problems with classmates and faculty is an excellent way to learn the material. Discussing homework problems with AI is a pretty good way to learn the material, but it will make you a little less happy. Be aware that it is easy to overestimate how much you actually understand individually when you solve problems with others.
Participation: Participation will be based on attendance and contributions during lecture. If you have personal circumstances that make it difficult for you to attend lecture, please contact me within the first two weeks of classes to make an alternative arrangement.
Technology in class: Laptops are permitted to be briefly opened for checking references, but should typically remain put away during lecture, unless you are one of the few people who can live tex your notes :). Phones should remain put away. iPads are permitted for notetaking. Only in the past year have I had to make rules about this, lol (:
Grading Scheme:
-
Your course grade will be determined by whichever of
the following two schemes gives you the higher score:
- Scheme A: Participation: 5%, Homework: 15%, Highest of Two Midterm Grades: 35%, Final: 45%
- Scheme B: Participation: 5%, Homework: 15%, Midterm 1: 25%, Midterm 2: 25%, Final: 30%
- All regrade requests must be received within two weeks after the graded work is returned.
- This is a core course for MATH and STSAP graduate students. Grades of A- or better will mean that you are performing at the Ph.D. level. Grades of B and B+ indicate performance at the MA level.
Prerequisites: undergraduate-level real analysis, similar to UCSB 118ABC
Outline of Course:
| Part I: Measures | Part II: Integration |
|---|---|
| sigma-algebras | measurable functions |
| measures | integration of functions |
| outer measures | modes of convergence |
| Lebesgue measure | product measures |
| Class | Date | topic | reading | due soon | notes |
|---|---|---|---|---|---|
| 1 | Sept 24 (Th) | introduction to measures | 1.1 | LEC1 | |
| 2 | Sept 29 (T) | sigma-algebras and measures | 1.2-1.3 | LEC2 | |
| 3 | Oct 1 (Th) | outer measures | 1.4 | HW1 HW1SOL | LEC3 |
| 4 | Oct 6 (T) | Borel measures on the real line (I) | 1.5 | ||
| 5 | Oct 8 (Th) | Borel measures on the real line (II) | HW2 | ||
| 6 | Oct 13 (T) | Borel measures on the real line (III) | |||
| 7 | Oct 15 (Th) | measurable functions | 2.1 | ||
| 8 | Oct 20 (T) | first midterm, over lectures 1-7 | |||
| 9 | Oct 22 (Th) | integration of nonnegative functions (I) | 2.2 | ||
| 10 | Oct 27 (T) | integration of nonnegative functions (II) | |||
| 11 | Oct 29 (Th) | integration of real-valued functions | 2.3 | ||
| 12 | Nov 3 (T) | modes of convergence (I) | 2.4 | ||
| 13 | Nov 5 (Th) | modes of convergence (II) | |||
| 14 | Nov 10 (T) | modes of convergence (III) | |||
| 15 | Nov 12 (Th) | second midterm, over lectures 1-14 | |||
| 16 | Nov 17 (T) | product measures (I) | 1.2, 2.5 | ||
| 17 | Nov 19 (Th) | product measures (II) | |||
| Nov 24 (T) | no class | ||||
| Nov 26 (Th) | no class - Thanksgiving holiday | ||||
| 18 | Dec 1 (T) | product measures (III) - Fubini-Tonelli | |||
| 19 | Dec 3 (Th) | n-dimensional Lebesgue measure | 2.6 | ||
| Dec 4 (F) | optional review session, time TBD | ||||
| Dec 9 (W) | final exam, 12-3pm |
Acknowledgements: I would like to thank Chuck Akemann and Davit Harutyunyan for sharing their materials from previous sessions of Math 201a at UCSB. I would also like to acknowledge Eric Carlen (Rutgers) and Brian White (Stanford), from whom I learned measure theory. I have referred to materials from their courses in preparing this one.