# MATH 621 (Spring 2019)

## Instructor's contact information

Michael Robinson
220 Don Myers Technology and Innovation Building
michaelr at american {dot} edu
Feel free to contact me with any and all questions (course-related or not)

Course description
Homework assignments
Course schedule
Course policies

# Course description

Measure Theory and Integration (3) This course presents the fundamental concepts and techniques of measure theory. It includes Borel sets, measures, measurable sets and functions, integrals as measures, Lp spaces, modes of convergence, and decomposition and generation of measures (including product measure). Crosslist: MATH-421. Usually Offered: alternate springs (even years). Prerequisite: MATH-603 and MATH-620.

In this course, students will

1. Learn the basics of Lebesgue measure in Euclidean space,
2. Learn the general theory of measurability, integration, and approximation
3. Learn some of the basic applications of these concepts.
There is no required book, however much of the spirit of the course is captured in two books:
• Frank Jones, Lebesgue integration on Euclidean space, Jones and Bartlett.
• Gerald Folland, Real analysis: Modern techniques and their applications, Wiley.

# Homework

Homeworks are on paper, and will be circulated in due time! They will mostly come from the Frank Jones book. Typesetting your answers in LaTeX is encouraged but not required.

Homework 1 due February 1
Homework 2 due February 22
Homework 3 due March 19
Homework 4 due April 2
Homework 5 due April 19
Homework 6 due April 30

# Course schedule

## Unit 1: Lebesgue measure on Euclidean space

After this unit, you should be able to
1. Describe which Euclidean subsets are measurable
2. Explain how to compute Lebesgue measure of open and compact Euclidean subsets

January 18: Construction of the Lebesgue measure
January 25: Properties of the Lebesgue measure
January 29: Dr. Robinson on travel

## Unit 2: Invariance of Lebesgue measure

After this unit, you should be able to
1. Compute Lebesgue measure of subsets that are affine transformations of others
2. Describe the measure of Cantor sets

February 1: Linear algebra of transformations
February 5: Translation and dilation
February 8: Ortogonal matrices
February 12: General matrix action on Lebesgue measure
February 15: Non-measurable sets exist
February 19: Cantor sets

## Unit 3: Algebras of sets and measurability

After this unit, you should be able to
1. Describe general conditions for measurability of sets and functions

February 22: Algebras and sigma algebras
February 26: Borel sets
March 1: No class
March 5: Non-Borel sets
March 8: Measurable functions, simple functions
March 12: Spring break
March 15: Spring break

## Unit 4: Lebesgue integration

After this unit, you should be able to
1. Compute the Lebesgue integral for some measurable functions
2. Describe useful theoretical properties of Lebesgue integration

March 19: Lebesgue integrals for nonnegative functions
March 22: Lebesgue integrals, generally
March 26: Almost everywhere
March 29: Integration in Euclidean space

## Unit 5: Lebesgue interal in Euclidean space

After this unit, you should be able to
1. Construct Lebesgue integrals for general measure spaces
2. Demonstrate conditions under which a sequence of L1 functions converges

April 2: Measure spaces
April 5: The Riemann integral
April 9: Changes of variables
April 12: Approximation of functions in L1
April 16: Continuity of translation L1

## Unit 6: Lp spaces

After this unit, you should be able to
1. Describe the Lp norm, and explain its usefulness in approximation

April 19: Normed spaces
April 23: Completeness
April 26: Review and final comments
Final exam: None planned

# Course policies

The course grade will be determined as the equal-weighted average of the homeworks.

Late homeworks will not be accepted, unless there is an official (University-approved) reason for doing so.

Academic dishonesty is a serious offense. As a start, you should read and understand our University's policies.

There is to be no collaboration with other people using any means during an exam. Doing so risks starting academic dishonesty proceedings. However, you may ask me questions during an exam, which I may answer at my discretion.

You may collaborate with other students in this class on homeworks, but the work you turn in must be your own. You may not collaborate with others outside of the class on homeworks without my express permission. This prohibition extends to the use of online forums and paid tutors. If you feel that you don't know how to proceed on an assignment, ask me for help!

At a minimum, honesty consists of presenting your ideas clearly and in your own words, possibly orally. On the other hand, the creation and writing of calculations, proofs, examples, or counterexamples is a creative process. Here are a few typical cases that are relevant for homework:

1. If you create a calculation, example, counterexample, or proof for your work on your own, you need not notate it as such. (It is expected that you will rediscover standard examples, and these will not cause any concern.)
2. If a colleague shows you an example, counterexample, or proof that you like, please credit that person by name in your write-up. You should expect that I may challenge you to explain your writing in your own words, possibly orally. If you cannot successfully defend your answer, no credit will be awarded to you.
3. If you write a calculation, example, counterexample, or proof as part of a group, please credit all members of your group by name in your write-up. You should expect that I may challenge you to explain your writing in your own words, possibly orally. If you cannot successfully defend your answer, no credit will be awarded to you, even if other group members are able to defend the same answer.

The mathematics community has a unique perspective on academic honesty and priority. The mathematics community has strict social guidelines for assigning credit, which I expect you'll adhere to. Due to the canonicity of certain mathematical results and constructions, sometimes well-intentioned people end up in priority disputes when they independently discover something. If you have any questions on this matter, you are expected to consult me directly for advice.