Instructor  Dr. Mark Comerford 
Office  Lippitt 102 F 
Phone  874 5984 
mcomerford@math.uri.edu  
Office Hours 
Monday 10am12pm or by appointment 
Text  H. L. Royden, P. M. Fitzpatrick Real Analysis (4th Ed.) ISBN 013143747X 
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 nonNegative Measurable Functions  
18.3 Integration of General Measurable Functions  5, 9 
18.4 The RadonNikodym Theorem  II, III Problems 
19.1 The Completeness of L^{p} (X, μ)  2, 27 (27 is extra credit) 
19.2 The Riesz Representation Theorem for the Dual of L^{p} (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 R^{n} 
Exams
Midterm  23:15pm Wednesday March 7 Lippitt 201 
Final  TBA Lippitt 201 
Evaluation
Homework  40% 
Midterm  20% 
Final  40% 
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:
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, 8742098.