# MATH 425: Integration Theory, Fall 2005

## Description:

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 Rn 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 Rn
• Approximation and continuity properties in Rn
• Fubini's Theorem
• Lp spaces

## Instructor:

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

## Text:

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%`

## Homework:

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.

Homework Assignments

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!

Midterm Exam Solutions

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.