640: 501 Theory of Functions of a Real Variable
Spring 02

Course Downloads

Lecture Notes for lecture 6
Lecture Notes for lectures 6 and 7
Problems 27-32
Lecture on algebras and sigma-algebras
Problems 33-44
Lecture notes on construction of measures
Lecture notes on Lebesgue-Stieltjes measures
Problems 45-52
Lecture notes; Approximation of Borel Measures
Problems 55-66
Problems 67-85 There are lots of problems here. No need to try all right away,but hand in 68, 69, 71, 79, 82 on Dec 1.
Notes on the proof of the Fubini-Tonelli Theorem

General Information

  1. Instructor: Daniel Ocone
  2. Class time and place: Hill 501, MW 1:10-2:30
  3. Current lecture topics and problem assignments Click on this link for a lecture by lecture summary of topics and current problem assignments.
  4. Problem sets Click here for PDF files of homework problems.

This course is an introduction to the theory of measure and integration. It covers basic material required for study of any area of mathematical analysis---ordinary and partial differential equations, harmonic analysis, probability and stochastic processes, complex analysis. A more detailed list of topics may be found in the course syllabus.

TEXT AND BOOKS ON RESERVE: The text for the course is:
Wheeden and Zygmund, Measure and Integeral: An Introduction to Real Analysis, Marcel Dekker, New York, 1977, ISBN 0-8247-6499-4.
This text is intended for reference and collateral reading. The lectures will not follow order of presentation of the book. Material that is not in the book or is taken out of context will sometimes be summarized in lecture notes, which will be available by links from this page. Also, there are several books on reserves in the Math Library in the Hill Center for this course, and students will be directed to these sources from time to time. These reserve books are:

  1. Folland, G., Real Analysis: Modern Techniques and their Applications, Wiley-Interscience.
  2. Halmos, P., Measure Theory, Van-Nostrand.
  3. Taylor, A.E., General Theory of Functions and Integration, Dover Books.
Folland's book is an excellent, modern text, which gives a very concise and organized account of essential real analysis techniques, but is more abstract than the course text. Folland's book also contains a brief introduction to general topology and material on ordinal numbers and transfinite induction. The Halmos book is a classic text, beautifully written, again from a fairly axiomatic and general viewpoint. The book by Taylor covers point set topology, as well as material on measure and integration, and includes a lot of examples. (Also, it is a Dover reprint, so you can buy it cheap if you like it!)

PREREQUISITES: It is assume that the student has a solid background in an undergraduate analysis course at the level of W. Rudin, Principles of Mathematical Analysis, and that the student can read and write mathematical proofs. In particular, the student should understand basic metric space topology---including compact sets, completeness, continuous functions and their properties---and the notion of countable set, the fact that the rationals are countable, but the reals are not, and similar basic facts. Summaries of basic points are contained in Chapter 1 of Wheeden and Zygmund and in sections P.5,P.6 of Folland.

COURSE WORK: This course emphasizes problem sets, which will be due weekly. Reading will be assigned for each class, and it is important that students prepare for class by doing the reading assignment. Current homeworks and reading assignments will be posted on a supplementary information page.