Math 201A: Real Analysis

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

** Lecture:** Tuesday and Thursday, 11am-12:15pm, Building 387, Room 1015

** Office Hours:** Monday and Friday 2-3pm and by appointment.

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

** 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 25th, 11am-12:15pm
- Second Midterm: Thursday, November 17th, 11am-12:15pm
- Final Exam: Wednesday, December 7, 12-3pm

** Homework:**

- Homework will be due Wednesdays 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:
- 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 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.

**Participation:** Participation will be based on attendance 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.

** Grading Scheme:**

- Participation: 5%, Homework: 30%, Highest of Two Midterm Grades 30%, Final 35%
- 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 |

topic | reading | due today | notes | ||
---|---|---|---|---|---|

1 | Sept 22 (Th) | introduction to measures | 1.1 | LEC1 | |

2 | Sept 27 (T) | sigma-algebras (I) | 1.2 | HW1, HW1SOL | LEC2 |

3 | Sept 29 (Th) | sigma-algebras (II) | LEC3 | ||

4 | Oct 4 (T) | measures and outer measures | 1.3-1.4 | LEC4 | |

5 | Oct 6 (Th) | Borel measures on the real line (I) | 1.5 | HW2, HW2SOL | LEC5 |

6 | Oct 11 (T) | Borel measures on the real line (II) | LEC6 | ||

7 | Oct 13 (Th) | measurable functions (I) | 2.1 | HW3, HW3SOL | LEC7 |

8 | Oct 18 (T) | measurable functions (II) | LEC8 | ||

9 | Oct 20 (Th) | integration of nonnegative functions (I) | 2.2 | HW4, HW4SOL | LEC9 |

10 | Oct 25 (T) | first midterm, over lectures 1-8 |
MID1, MID1SOL | ||

11 | Oct 27 (Th) | integration of nonnegative functions (II) | LEC10 | ||

12 | Nov 1 (T) | integration of real valued functions | 2.3 | LEC11 | |

13 | Nov 3 (Th) | modes of convergence (I) | 2.4 | HW5, HW5SOL | LEC12 |

14 | Nov 8 (T) | modes of convergence (II) | LEC13 | ||

15 | Nov 10 (Th) | modes of convergence (III) | HW6, HW6SOL | LEC14 | |

16 | Nov 15 (T) | product measures (I) | 2.5 | LEC15 | |

17 | Nov 17 (Th) | second midterm, over lectures 1-14 |
MID2_Prac, MID2, MID2SOL | ||

18 | Nov 22 (T) | very serious mathematics ( special online lecture) |
HW7, HW7SOL | ||

19 | Nov 29 (T) | product measures (II) | LEC16 | ||

20 | Dec 1 (Th) | Fubini-Tonelli | HW8, HW8SOL | LEC17 | |

Dec 7 (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 the materials from their courses in preparing this one.