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


Weekly Plan:
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.