Math 4110/6110: the Lebesgue integral Joe Fu

Spring 2006 407 Boyd

fu@math.uga.edu

542-2564

Overview of the course: Our aim in this course is, naturally, to introduce you to the basic theory of Lebesgue integration. The theory dates from the early years of the 20^th century, and represents a definitive solution to the problem of understanding what the integral really is and how generally it can be applied (e.g. the “improper integrals” that must be treated as special cases in the Riemann theory need no such handling here). As a result, the Lebesgue integral has served as the foundation for the explosion of knowledge in analysis since then.

For example. one area in analysis that has been completely transformed--- one might even say created--- by the advent of the Lebesgue integral is Probability Theory. (From another perspective one could say, with only slight exaggeration, that each includes the other.) Our text, Measure theory and integration, by Victor Guillemin and our very own Malcolm Adams, is constructed around this nexus. We will follow this line rather closely, with some excursions into more geometric applications. I expect we will cover the first two chapters completely, together with a good chunk of Chapter 3.

Homework: Problem sets will be assigned roughly every other week. The problems will come in two varieties. The ones decorated with *s will be (in my view) the more challenging ones, and sometimes a bit open-ended. These may be handed in at any time before the final exam for full credit. The rest must be handed in by the due date unless special arrangements have been made with me and with the grader.

You may consult freely with other students and with other sources (books, internet, etc.) on all problem sets. If so you must cite these people and sources clearly in your paper, identifying what help you have received on which problems. Furthermore, everything you hand in must reflect your own actual understanding--- don’t just copy your friend’s solutions. I may quiz you orally in class from time to time to verify this.

The grader for the course is Chao Zhuang (czhuang@math.uga.edu).

Exams: There will be an in-class mid-term exam, tentatively scheduled for Friday, March 3. The final exam is scheduled for Friday, May 5, 12 noon—3 pm. However I may substitute a take-home exam for this. The ground rules for the take-home exams will be different from problem sets: no collaborations will be permitted. We’ll discuss this further as the date approaches.

Grading policy: Grades will be assigned according to the following weights:

Homework: 40%

mid-term: 20%

final: 30%

participation and effort: 10%

Academic honesty: All academic work must meet the standards contained in “A Culture of Honesty.” Students are responsible for informing themselves about those standards before performing any academic work.