``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. This course is intended for students in Mathematics, Statistics, Engineering and other sciences.

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. We will emphasize throughout the lectures examples steaming from probability theory, every student in measure theory should be acquainted with the fundamental concepts and functions specific to this part. The dictionary that allows us to move from measure theory to probability theory will be laid out so as to help travelers in either direction to feel comfortable (eg: random variable in probabilistic jargon corresponds to measurable function in measure theory).

Texts:The book we will use is The Elements of Integration and Lebesgue Measure by Robert G. Bartle, Wiley Classics Library, 1995. A couple of books are recommended: Real Analysis: Modern Techniques and their Applications by Gerald Folland, John Wiley and Sons, 1999 (2nd 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. There are many classical books on Measure Theory or Real Analysis available, such as Royden's Real Analysis and Rudin's Real and Complex Analysis most of them will contain all of the above and more, they are also excellent references for this course.

Grades: Grades will be based on homeworks and/or projects.

Prerequisites: Real Analysis (MATH 510 or MATH 361) or permission from the instructor.

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

Last updated: 6 August, 1999