Math 201A: Real Analysis

**Professor:** Katy Craig, katy•craig at ucsb • edu

** Lecture:** Tuesday and Thursday, 9:30-10:45am, zoom (recordings will be posted below)

** Office Hours:** Monday 10-11am and 4:30-5:30pm, and by appointment, zoom

** Wiki: ** Measure Theory Wiki

** Textbook: ** Folland, * Real Analysis: Modern Techniques and Their Applications*, second edition

the library will scan the first and second chapters of this textbook for student use: course reserves

** Other Recommended References:**

** 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, as the grading scheme allows you to drop your lowest midterm score.

- First Midterm: Tuesday, October 27th, 9:30-10:45AM
- Second Midterm: Tuesday, November 24th, 9:30-10:45AM
- Final Exam: Tuesday, December 15th, 8:00-11:00AM

** Homework:**

- Homework will be due Tuesdays at 9:30pm.
- Assignments will be posted on this website and submitted via Gradescope; see Discord for instructions.

- 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:
- 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.
- Discussing homework problems with classmates is an excellent way to learn the material. However, be aware that it's easy to overestimate how much you actually understand when you work in a group.

**Wiki Assignment:**

- Each student will be required to write one short wiki article and edit a classmate's wiki article.
- Writing/editing extra articles is highly encouraged and will probably help you learn more.
- Here is a short video explaining how to create a new article on the wiki.

**Participation:** Participation will be based on attendance during synchronous zoom lectures. If you have personal circumstances (such as time zone issues) that make it difficult for you to attend the synchronous lectures, please contact me within the first two weeks of classes to make an alternative arrangement.

** Grading Scheme:**

- Attendance: 5%, Wiki: 5%, Homework: 30%, Highest of Two Midterm Grades 30%, 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, but not quite at the Ph.D. 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 |

topic | reading | due today | notes/review materials | ||
---|---|---|---|---|---|

1 | Oct 1 (Th) | introduction to measures | 1.1 | LEC1, VID1, OHR1 | |

2 | Oct 6 (T) | sigma-algebras, part 1 | 1.2 | HW1, HW1SOL | LEC2, VID2 |

3 | Oct 8 (Th) | sigma-algebras, part 2 | LEC3, VID3, OHR2, OHR3 | ||

4 | Oct 13 (T) | measures, part 1 | 1.3 | HW2, HW2SOL | LEC4, VID4 |

5 | Oct 15 (Th) | measures, part 2 | LEC5, VID5, OHR4 OHR5 | ||

6 | Oct 20 (T) | outer measures, part 1 | 1.4 | HW3, HW3tex, | LEC6, VID6 |

7 | Oct 22 (Th) | outer measures, part 2 | |||

8 | Oct 27 (T) | first midterm, over lectures 1-7 |
|||

9 | Oct 29 (Th) | Borel measures on the real line, part 1 | 1.5 | ||

10 | Nov 3 (T) | Borel measures on the real line, part 2 | |||

11 | Nov 5 (Th) | measurable functions, part 1 | 2.1 | ||

12 | Nov 10 (T) | measurable functions, part 2 | |||

13 | Nov 12 (Th) | integration of nonnegative functions | 2.2 | ||

14 | Nov 17 (T) | integration of complex functions, part 1 | 2.3 | ||

15 | Nov 19 (T) | integration of complex functions, part 2 | |||

16 | Nov 24 (T) | second midterm, over lectures 8-16 |
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17 | Dec 1 (T) | modes of convergence, part 1 | 2.4 | ||

18 | Dec 3 (Th) | modes of convergence, part 2 | |||

19 | Dec 8 (T) | product measures, part 1 | |||

20 | Dec 10 (Th) | product measures, part 2 | |||

Dec 15 (T) | final exam, 8:00-11:00AM |

**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 the materials from their courses in preparing this one.