Math 201A: Real Analysis

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

Lecture: Tuesday and Thursday, 11am-12:15pm, Girvitz, Room 1115

Office Hours: Monday 12-1pm, Thursday 12:15-1:15pm in SH 6507

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: Thursday, October 24th, 11am-12:15pm
Second Midterm: Thursday, November 21st, 11am-12:15pm
Final Exam: Wednesday, December 11, 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 is an excellent way to learn the material. However, be aware that it's easy to overestimate how much you actually understand individually when you solve problems in a group.

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.

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 soon notes

1 Sept 26 (Th) introduction to measures 1.1 LEC1

2 Oct 1 (T) sigma-algebras and measures 1.2-1.3 LEC2

3 Oct 3 (Th) outer measures 1.4 HW1 HW1SOL LEC3

4 Oct 8 (T) Borel measures on the real line (I) 1.5 LEC4

5 Oct 10 (Th) Borel measures on the real line (II) HW2, HW2tex , HW2SOL LEC5

6 Oct 15 (T) Borel measures on the real line (III) LEC6

7 Oct 17 (Th) measurable functions 2.1 HW3, HW3tex , HW3SOL LEC7

8 Oct 22 (T) integration of nonnegative functions (I) 2.2 LEC8

9 Oct 24 (Th) first midterm, over lectures 1-7 PracMid1 Mid1 , Mid1SOL

10 Oct 29 (T) integration of nonnegative functions (II) LEC9

11 Oct 31 (Th) integration of real valued functions 2.3 HW4, HW4tex HW4SOL LEC10

12 Nov 5 (T) modes of convergence (I) 2.4 LEC11

13 Nov 7 (Th) modes of convergence (II) HW5, HW5tex , HW5SOL LEC12

14 Nov 12 (T) modes of convergence (III) LEC13

15 Nov 14 (Th) product measures (I) 1.2, 2.5 HW6, HW6tex , HW6SOL LEC14

16 Nov 19 (T) product measures (II) LEC15

17 Nov 21 (Th) second midterm, over lectures 1-14 PracMid2, MID2, MID2SOL

18 Nov 26 (T) product measures (III) - Fubini-Tonelli LEC16 LEC16_VIDEO

19 Nov 27 (W) n-dimensional Lebesgue measure 2.6 LEC17, LEC17_VIDEO

Dec 9 (M) optional review session, 11am-12:15pm HW7 , HW7SOL

Dec 11 (W) final exam, 12-3pm

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 soon	notes
1	Sept 26 (Th)	introduction to measures	1.1		LEC1
2	Oct 1 (T)	sigma-algebras and measures	1.2-1.3		LEC2
3	Oct 3 (Th)	outer measures	1.4	HW1 HW1SOL	LEC3
4	Oct 8 (T)	Borel measures on the real line (I)	1.5		LEC4
5	Oct 10 (Th)	Borel measures on the real line (II)		HW2, HW2tex , HW2SOL	LEC5
6	Oct 15 (T)	Borel measures on the real line (III)			LEC6
7	Oct 17 (Th)	measurable functions	2.1	HW3, HW3tex , HW3SOL	LEC7
8	Oct 22 (T)	integration of nonnegative functions (I)	2.2		LEC8
9	Oct 24 (Th)	first midterm, over lectures 1-7		PracMid1 Mid1 , Mid1SOL
10	Oct 29 (T)	integration of nonnegative functions (II)			LEC9
11	Oct 31 (Th)	integration of real valued functions	2.3	HW4, HW4tex HW4SOL	LEC10
12	Nov 5 (T)	modes of convergence (I)	2.4		LEC11
13	Nov 7 (Th)	modes of convergence (II)		HW5, HW5tex , HW5SOL	LEC12
14	Nov 12 (T)	modes of convergence (III)			LEC13
15	Nov 14 (Th)	product measures (I)	1.2, 2.5	HW6, HW6tex , HW6SOL	LEC14
16	Nov 19 (T)	product measures (II)			LEC15
17	Nov 21 (Th)	second midterm, over lectures 1-14		PracMid2, MID2, MID2SOL
18	Nov 26 (T)	product measures (III) - Fubini-Tonelli			LEC16 LEC16_VIDEO
19	Nov 27 (W)	n-dimensional Lebesgue measure	2.6		LEC17, LEC17_VIDEO
	Dec 9 (M)	optional review session, 11am-12:15pm		HW7 , HW7SOL
	Dec 11 (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.