### MATH 6337 Real Analysis I

### Spring 2014

### TTh 12-1:30 Skiles 170

### Office Hours: TTh 2:00-3:00 in Skiles 224

If you have problem to meet me during the scheduled or
email me to set an appointment.

#### Prerequisite

MATH 4317, 4318 (Analysis I, II)

#### Textbook

Gerald B. Folland, "Real Analysis", Wiley Inter-Science, 2nd Edition.

Will cover chapter 1, 2, and 3 from the book plus other argument using prof. Heil notes and other material.

#### Material available on line

Professor C. Heil material for Math6337.

#### Grading

I will collect HW every 2 weeks for a total of 5/6 HW. On top there will be two in class midterms and a in
class final. Midterms and final will be cumulative. The possibility of personal project will be discussed
later during the class. The grading scheme will be 25% HW, 35% midterms and 40% final.

**First Midterm: **February 18 Second Midterm: March 25

**First week**

- Material covered
- 0.1-0.6 Prologue
- 1.1 Measure: introduction
- 1.2 Sigma Algebras

- Exercises

**Second week**

- Material covered
- Exercises

**First HW collection on Tuesday January 21nd. Solution set for the first HW.**

**Third week**

- Material covered
- (1.4) Outer Measure: Carathéodory's Theorem

- (1.5) Borel Measure on the Real Line

- Exercises
- (1.4) 18, 19

- (1.5) 28, 32, 33

**Fourth week**

- Material covered
- (1.5) Borel Measure on the Real Line (continued)

- Exercises

**Second HW collection on Thursday February 6th.**

**Fifth week**

- Material covered
- (2.1) Measurable Functions

- (2.2) Integration of Nonnegative Functions

- Exercises

**Seventh week**

- Material covered
- (2.3) Integration of Nonnegative Functions

- (2.4) Mode of Convergence

- Exercises
- (2.3) 19, 23, 27
- (2.4) 37, 43

First midterm on February 25th.

Practice material 1 and 2 for the Midterm.

Solution set for the first midter.

**Third HW collection on Tuesday March 11th.**

**Eighth week**

- Material covered
- (2.5) Product Measures
- (2.6) The n-dimensional Lebesgue Integral
- (2.7) Integration in Polar Coordinates

- Exercises
- (2.5) 47, 50
- (2.6) 56, 59
- (2.7) 64

**
Fourth HW collection on Teusday April 1st.**

Tenth week

- Material covered
- (3.1) Signed Measures
- (3.2) The Lebesgue-Radon-Nikodym Theorem

- Exercises
- (3.1) 3, 6, 7
- (3.2) 8, 12, 17

**Last HW collection on Tuesday April 15th**

Solution set for the second midter.

Take home midterm due on April 22.