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.

**First week**

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

- Exercises
- (1.2) 1, 4, 5

**Second week**

- Material covered
- 1.3 Measures

- Exercises
- (1.3) 8, 11, 16

**First HW collection on Tuesday January 22nd.**

Solution set** **for the first HW.

**Third week**

- Material covered
- Notes: Outer measure in Rn
- Notes: Outer measure.
- Notes: Lebesgue Measurable set

- Exercises

**Forth week**

- Material covered
- Notes: Lebesgue measure.
- (1.4) Outer Measure: Carathéodory's Theorem

- Exercises

**Second HW collection on Thursday February 7th.**

** **
Solution set** **for the second HW.

**Fifth week**

- Material covered
- (1.4) Outer Measure: Carathéodory's Theorem (Continued)
- (1.5) Borel Measure on the Real Line
- Exercises
- (1.4) Text: exercises n. 3
- (1.5) 28, 32, 33

- Material covered
- (2.1) Measurable function
- (2.2) Integration of Non-Negative Functions

- Exercises
- (2.1) 3, 4, 7
- (2.2) 13

Practice material 1 and 2 for the Midterm.

First midterm on February 21st. Here is the text and the solution set for the midterm.

Solution set- Material covered
- (2.3) Integration of Complex Functions
- (2.4) Modes of Convergence

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

**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

**Forth HW collection on Thursday March 27th.** Solution set** **for the forth HW.

Practice material 2 for the Midterm.

Second midterm on Tuesday April 1st. Here is the text and the solution set for the midterm. Exercise 5 of the test contained an error: to apply Egoroff theorem you need a.e. convergence that does not follow from the L1 convergence in the hypotheses of the exercise. This has been taken into account in the grading process.

**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 22th**