__Announcements:__

**Homework Assignments.****
**Exercises given after the description of each lecture below are suggested exercises
to do on your own. Of course I will give hints and will answer questions
in class or out of class about any of them. A subset of each set of these
exercises will be collected according to the schedule given below the lecture
descriptions. I will ask that these exercise solutions be typed up using
some kind of mathematical software, preferably TeX of
some flavor. Any software capable of reproducing mathematical symbols is
acceptable. Let me know if you have any questions about this.

**Deadlines.**** **Please
be aware of all relevant deadlines

__Class Notes
and Exercises.__

Lecture 1 This lecture covers certain basic facts from undergraduate Real Analysis, namely basic definitions, the Heine-Borel Theorem, the Nested Set Theorem and their equivalence to the Completeness Axiom for the real numbers. Exercises: Chapter 1, 3, 6, 12-14, 17, 23, 25.

Lecture 2 Sigma Algebras and Borel Sets. Sigma algebras and in particular the sigma algebra of the Borel sets are defined. Problem 44 of Section 1.5 is worked through in anticipation of the definition of the Cantor set later on. Exercises: Chpater 1, 30-32, 35-39, 43, 47, 48, 50, 51, 56, 57.

Lecture 3 Outer Measure and Measurability. We define the basic properties we require of a measure, then define outer measure and prove some of its basic properties. We then define measurability and show that the collection of measurable sets is a sigma algebra. Chapter 2, 6-10, 14, 15.

Lecture 4 Alternate Characterization of Measurability. We give an alternae characterization of measurability based on the approximability of sets by open sets from the outside and approximability by closed sets from the inside. We then define Lebesgue measure and show that it is countably additive and that it possesses a continuity property. Exercises: Chapter 2, 17, 18, 20, 21, 25, 26, 28.

__Lecture 5__ Nonmeasurable Sets, the Cantor set and the Cantor-Lebesgue function. These notes follow essentially
Sections 2.5 and 2.6 in the book. Exercises: Chapter 2, 29, 30, 33,
37, 38, 40, 44, 46.

__Lecture 6__ Measurable
Functions. Definition and basic properties of
measurable functions. These notes follow essentially the content
of Sections 3.1 and 3.2. Exercises: 1, 3-8, 10, 12, 14-16, 21.

__Lecture __7 Littlewood's
Three Principles. These notes follow Section 3.3 in the text.
Exercises: 27-31.

__Lecture 8__ Lebesgue Integral.
These notes follow Sections 4.1 and 4.2 in the text. Exercises: 3,
4, 5, 6, 8, 9, 12, 16.

__Lecture __9 Convergence Theorems.
These notes cover topics from Sections 4.3-4.6 in the text.
Exercises: 17, 19, 20, 22, 27, 28, 30, 32, 33, 38, 43, 44.

__Lecture 10 Vitali's
Convergence Theorems.__ These notes cover topics
from Sections 4.6 and 5.1 in the text. Exercises: Chapter 4; 50,
51, 52, Chapter 5; 1, 4, 5.

__Lecture 11__ Convergence
in Measure. These notes cover topics from Sections 5.2, and 5.3 in the
text. Exercises: 11, 12, 13, 14.

__Lecture 12__
The Fundamental Theorem of Calculus, Part 1.
Exercises are given below under Homework #8.

__Lecture 13__ The Vitali Covering Lemma and the Lebesgue
Differentiation Theorem. These notes
cover Sections 6.1 and 6.2 in the text.
Exercises: 9, 10, 12, 13, 15, 24.

__Lecture 14__
The Fundamental Theorem of Calculus, Part 2.
These notes cover Sections 6.3-6.5 in the text. Exercises:
26, 27, 29, 33, 35, 38, 39, 40, 52, 55, 60.

__Lecture 15__
General Measure Spaces. These notes
cover Sections 17.1 and 17.2 in the text.
Exercises: 1, 2, 4, 7, 8, 12, 13,
14, 16.

__Lecture 16__ Integration
on General Measure Spaces and the Radon-Nikodym
Theorem. These notes cover
Sections 18.1-18.4 in the text.
Exercises: 2, 3, 18, 19, 25, 26,
44, 45, 49, 52, 53, 54, 60.

__Lecture 17__ The Theorems of Fubini and Tonelli. These notes cover Sections 20.1 and 20.2 in
the text. Exercises: 5, 6, 10, 11, 12.

**Homework Assignments:**

**Homework #1
(due 07 February): **Chapter 1, Exercises 3, 17, 23, 36, 38, 39,
50, 51.

**Homework #2
(due 14 February): **Chapter 2, Exercises 14, 15, 20, 21, 26,
29, 33.

**Homework #3
(due 21 February): **Chapter 2, Exercises 38, 44. Chapter
3, Exercises 5-8 .

**Homework #4
(due 28 February): **Chapter 3, Exercises 14-16, 27, 28, 31.

**Homework #5
(due 07 March): **Chapter 4, Exercises
4, 5, 12, 16.

**Homework #6
(due 21 March): **Chapter 4, Exercises
17, 22, 27, 30, 33, 38, 44.

**Homework #7
(due 28 March): **Chapter 4, Exercise
52, Chapter 5, Exercises 1, 11, 13, 14.

**Homework #8
(due 11 April): 776s11HW8.pdf**

**Homework #9
(due 18 April): **Chapter 6, Exercises 29, 35, 39, 40, 55, 60.

**Homework #10
(due 25 April):
**Chapter 17, 1, 2, 8, 13, 16.**
**

**Homework #11
(due 02 May): **Chapter 18, 25, 26, 44, 45, 52, 54.**
**

**Homework #12 (due 09
May): **Chapter 20, 5, 6, 12.

To contact me, send mail to: *dwalnut@gmu.edu*.