220 Don Myers Technology and Innovation Building

michaelr at american {dot} edu

Office hours: by appointment (please contact me 24 hours in advance to make arrangements). My research website http://www.drmichaelrobinson.net/

Feel free to contact me with any and all questions (course-related or not)

Homework assignments

Course schedule

Information about exams

Course policies

In this course, students will

- Learn the basics of Lebesgue measure in Euclidean space,
- Learn the general theory of measurability, integration, and approximation
- Learn some of the basic applications of these concepts.

- Frank Jones,
*Lebesgue integration on Euclidean space*, Jones and Bartlett. - Gerald Folland,
*Real analysis: Modern techniques and their applications*, Wiley.

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

- Describe which Euclidean subsets are measurable
- Explain how to compute Lebesgue measure of open and compact Euclidean subsets

January 18: Construction of the Lebesgue measure

January 22: General additivity

January 25: Properties of the Lebesgue measure

January 29: Dr. Robinson on travel

- Compute Lebesgue measure of subsets that are affine transformations of others
- 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

- 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

- Compute the Lebesgue integral for some measurable functions
- 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

- Construct Lebesgue integrals for general measure spaces
- 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

- 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 **

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

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:

- 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.)
- 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.
- 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.