Approximately 100 years ago, Henri
Lebesgue described a new theory of integration to
handle many important, possibly discontinuous, functions of several
variables. This discovery has proven to be one of the most influential
mathematical events of history. For example it greatly advanced the development
of probability theory and Fourier analysis. Remarkably, though enjoying
many extensions and generalizations, it retains its original formulation while
in actve use today. In this course, we will
carefully develop *Lebesgue's** integration
theory*. This applies to quite general measures, though we will focus on the
important *Lebesgue** measure* on R^{n} which roughly describes n dimensional volume. Some of the topics
to be covered are as follows:

- Elementary topologicl notions in Euclidean space.
- Construction and some properties of Lebesgue measure
- Invariance under rigid motion
- Lebesgue measurability of sets
- Cantor set and the Lebesgue-Cantor function
- Sigma algebras and Borel sets
- General measures and Lebesgue integration
- Riemann and Lebesgue integrals in R
^{n} - Approximation and continuity
properties in R
^{n} - Fubini's Theorem
- L
^{p}spaces

Robert Hardt

Office: HB 430; Office hours: 9-10 MWF

(and others by appt.)

Excluded times:M:10-1,3-5,
T:10-1:30,4-5, W:10-1,1-2,3-5, Th:10-1:30,4-5, F:10-11,3-4

Email: hardt@rice.edu

Telephone: ext 3280

Frank Jones, *Lebesgue**
Integration on Euclidean Space*, Jones and Bartlett Publishers Inc., 1993. (available from the Rice bookstore or Dr. Jones
(HB448). This resource gives an excellent careful treatment and is a
great source of exercises. In Math 425 we will cover approximately
1/2 of this text. Some other commonly used texts are Bartle,
*Elements of Integration and Lebesgue
Measure,* Royden, *Real Analysis,* Rudin, *Real and Complex Analysis,* and and Wheeden and Zygmund, *Measure
and Integral.*

The final grade for the course will be determined as follows:

Homework 35%

Midterm exam 25%

Final exam 40%

There will be a homework assignment each week when there is not an hour exam scheduled. All homework is usually due each Wednesday, one week after it is assigned. In doing any problem in the text you may find it useful to use the results of previous problems in the text. The homework is not pledged. You are encouraged to discuss the homework, and to work together on the problems. However each student is responsible for the final preparation of his or her own homework papers.

**Note
the changes in Homework 4.**

**Note
Homework 5 now due Monday Oct. 3 because of Rita.**

**Note
the change in Problem 8 of Homework 8.**

**Midterm Exam.****
**This is a take-home exam
that will be distributed in sealed envelopes Monday Oct.17. It will be due back Monday Oct.24 at
10AM. After opening the envelope
you should work for no more than the next 2 hours. You may use the textbook or notes. Be
sure and hand in all the work you do on each problem. Good luck!

**Final Exam**.
This is a take-home final that will be distributed in sealed envelopes
Friday December 2. It will be due
back at the end of Finals period Wednesday Dec.14, 4PM. After opening the envelope you should
work for no more than the next 4 hours.
You may use the textbook or notes or old homeworks.
Be sure and hand in all the work you do on each problem. Good luck!

The final contains a few homework problems
and a few problems similar to those on the midterm exam.

The last homework #11 is graded and outside
the door of HB430.

This
page is maintained by Robert Hardt, hardt@rice.edu.

Last edited 12/6/05.