*``Towards the end of the nineteenth century it became clear to many
mathematicians that the Riemann integral
(about which one learns in calculus courses) should be replaced by some other
type of integral, more general and more flexible, better suited for dealing
with limiting processes. Among the attempts made in this direction, the most
notable ones were due to Jordan, Borel, W. H. Young, and Lebesgue.
It was Lebesgue's construction which turned out to be the most
succesful.''** The theory
continued to develop until the early 50's when it assumed more or less the
form in which we know it today.
As some authors phrase it, restricting ourselves to
Riemann integrable functions was like working
with the rational numbers without acknowledging the existence of irrational
numbers.
Abstract measure and integration theory is a far-reaching and beautiful
piece of mathematics that
should be part of the general mathematical
culture any graduate student in mathematics or statistics is exposed to.
This course is an introduction to Lebesgue Integration and Measure Theory which
extends familiar notions of length, volume, integration to
more general settings. Mathematical probability is an important part of
measure theory, this course should provide an excellent background for an
advanced course in probability. It is also fundamental background for advanced
courses in Functional Analysis, Differential Equations, Harmonic Analysis.

** Topics will include:** Measurable sets and functions, measures
and measure spaces (in particular Lebesgue measure). Next we will develop and
integration theory that generalizes Riemann's Integral, and prove basic
convergence theorems (Fatou's Lemma, Monotone Convergence Theorem,
Lebesgue Dominated Convergence Theorem).
Lebesgue spaces ($L^p$) and basic inequalities will be presented.
Different modes of convergence will be introduced and compared.
We will discuss decomposition and differentiation of measures,
and an excursion into
functions of bounded variation and absolutely continuous functions is in
order. Product measures will be studied and the celebrated Fubini's theorem
(on the interchange of integrals) will be proved.
Time permitting we will study Hausdorff measure and fractals.

** Texts:** The book we will use is *Real Analysis* by Elias
Stein and Rami Shakarchi. Princeton Lectures in Analysis III.
Princeton Press 2005. We will also use Terence Tao * Introduction to Measure Theory* which exists as a book published by the AMS and is also available online as a pdf, and annotations and errata can be found in Tao's blog
(which is very interesting)
**Tao's blog**.

We also recommend the text
* The Elements of Integration and
Lebesgue Measure* by Robert G. Bartle, Wiley Classics Library, 1995.
There are a number of classic books that cover the material we will study
and more, and that you might want to explore,
* Real Analysis: Modern Techniques and their Applications* by
Gerald Folland, John Wiley and Sons, 1999 (2nd edition);
*Real Analysis* by Halsey Royden, Prentice Hall, 1988
(3rd Edition)**;
*Real and Complex Analysis* by Walter Rudin, McGraw-Hill
Science/Engineering/Math, 1987 (3rd revised edition)*; and
* Measure Theory* by J. L. Doob, Springer-Verlag, GTM 144, 1994.
The later book
is written by a probabilist, it integrates probabilistic concepts in the text,
and many examples are taken from probability theory.
The above are all excellent references for this course.

**Prerequisites:** Real Analysis
**MATH 510** or
** MATH 401/501** or permission
from the instructor.

** Americans with Disabilities Act:**
Qualified students with disabilities needing appropriate academic
adjustments should contact me as soon as possible to ensure your needs are
met in a timely manner. Handouts are available in alternative accessible
formats upon request.

** 4th Edition co-authored with Patrick Fitzpatrick. Prentice Hall (2007).

Return to: Department of Mathematics and Statistics, University of New Mexico

Last updated: August 6, 2014