University of Rhode Island      Department of Mathematics

MTH 536  Measure and Integration II

M, W   2-3:15pm Lippitt Hall 201

 Instructor Dr. Mark Comerford Office Lippitt 102 F Phone 874 5984 Email mcomerford@math.uri.edu Office Hours Monday 10am-12pmor by appointment Text H. L. Royden, P. M. Fitzpatrick Real Analysis (4th Ed.) ISBN 0-13-143747-X Prerequisites Mth 535, 435, 436 or equivalent

Lectures and Homework Problems (Tentative)

 Section Problems 4.6 Uniform Integrability: The Vitali Convergence       Theorem 5.1 Uniform Integrability and Tightness 5.2 Convergence in Measure 5.3 Characterizations of Riemann and Lebesgue       Integrability 6.1 Continuity of Monotone Functions 6.2 Differentiability of Monotone Functions:       Lebesgue's Theorem 6.3 Functions of Bounded Variation:       Jordan's Theorem 6.4 Absolutely Continuous Functions 6.5 The Fundamental Theorem of Calculus 6.6 Convex Functions 17.1 Measures and Measurable Sets 17.2 Signed Measures: the Hahn and Jordan         Decomposition Theorems 17.3 The Carathéodory Measure Induced by an        Outer Measure 18.1 Measurable Functions 9, 5, 8 18.2 Integration of non-Negative Measurable         Functions 18.3 Integration of General Measurable Functions 5, 9 18.4 The Radon-Nikodym Theorem II, III Problems 19.1 The Completeness of Lp (X, μ) 2, 27 (27 is extra credit) 19.2 The Riesz Representation Theorem for the         Dual of Lp (X, μ), 1 ≤ p < ∞ 12 19.3 The Kantorovich Representation Theorem for         the Dual of L∞ (X, μ) 12 20.1 Product Measures: the Theorems of Fubini and         Tonelli 18 20.2 Lebesgue Measure on Rn

Exams

 Midterm 2-3:15pm Wednesday March 7  Lippitt 201 Final TBA   Lippitt 201

Evaluation

 Homework 40% Midterm 20% Final 40%

A 95 - 100, A- 90 - 95, B+ 87 - 90, B 83 - 87, B- 80 - 83, C+ 77 - 80, C 73 - 77, C- 70 - 73, D+ 67 - 70, D 60 - 67, F < 60.

Course Description

This course is an introduction to the wonderful world of measure and integration. You will learn how it is possible to give meaningful answers for the integrals of functions whose Riemann integrals do not exist. Key to this is the idea of a measure, which is a way of giving the size of sets of real numbers which are much more general than the simple intervals used in the Riemann integral. We will rigorously explore questions such as which functions can be integrated in this way, when and where can one differentiate `nice' functions such as monotone functions, when can one interchange the order of integration and when can one measure can be obtained by integrating a density against another measure.

In addition to covering the course material, there will be a very strong emphasis on mathematical rigor and mastering techniques of proof. This is reflected in the rather large percentage of the overall grade which is given over to homework which will be assigned weekly. Please note two things about the homework: firstly, I expect all homework to be submitted using LaTEX and secondly that I will not be taking resubmissions.

In addition to homework, there will be one midterm (provisional date Wednesday October 21) and a final. The material covered in the exams will be partly based on the homework, but there is also some expectation that there will be unseen problems. In addition, you may be asked to give proofs of theorems covered in class and you will be given in advance a list of those theorems which are examinable.

Goals and Objectives

The goals of the course are to have you develop the skills of working with real numbers, limits, continuity and differentiability.

At the conclusion of this semester you should be able to:

1. Understand what a sigma algebra of sets is.

2. Understand the construction of Lebesgue measure on the real line by means of outer measure.

3. Understand the Lebesgue integral, when it can be used and how it extends the definition of the Riemann integral.

4. Understand and work with the concept of differentiability of functions which are monotone, of bounded variation, or absolutely continuous.

5. Be able to read mathematics and construct a rigorous mathematical argument.

Special Accommodations

Students who need special accomodations and who have documentation from Disability Services should make arrangements with me as soon as possible. Students should conact Disability Services for Students, Office of Student Life, 330 Memorial Union, 874-2098.