### MATH 776-001 - Measure and Integral - Spring 2011

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.

Course syllabus pdf,   html

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.

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

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.

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.

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.

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.

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.