# Math 563, Measure Theory, Fall 2009

## General Information

Instructor: Matthew Blair
Course Web Page: http://www.math.unm.edu/~blair/math563.html
Office: Humanities 443
Office Hours: Wednesday 3:00-5:00 pm, and by appointment.

Text: Real Analysis: Measure Theory, Integration, and Hilbert Spaces by Elias M. Stein and Rami Shakarchi, Princeton University Press, 2005. It is expected that we will cover the main topics in Chapters 1, 2, 3, and 6.

The website for the text, along with a PDF of Chapter 1 can be found here.

Meeting times/location: Monday, Wednesday, and Friday at 2 pm, Humanities 424.

Prerequisites: Real analysis at the level of Math 510 or 401/501, including a firm grasp of point set topology and the Riemann integral. Consult the instructor with any questions.

Course Description

Towards the end of the nineteenth century, it was realized that the Riemann integral was not robust enough to handle all the needed applications. For example, Riemann's theory does not deal with convergence issues particularly well, such as "passing the limit under the integral sign". However, this is important for many applications of integration, including developing a rigorous foundation for Fourier analysis. A successful alternative was discovered by Henri Lebesgue in his 1902 thesis, who proposed to define the integral by more or less partitioning the range instead of the domain. In subsequent years this notion of integration has seen significant refinements and generalization.

In this course, we will begin by defining Lebesgue measure on Euclidean space and use this to integrate functions. We will then explore several important consequences of the Lebesgue integral such as convergence, differentiation, and averaging theorems. Finally, we will study abstract measure theory, where one can begin to consider notions of measure and integration over sets which may not be Euclidean in nature. If time permits, we will also discuss Lp spaces. This course will provide a solid (and often essential) foundation for students interested in differential equations, probability, functional analysis, and harmonic analysis.

Homework

Assignment #1--Due Monday, September 14

Stein and Shakarchi: Exercises 14, 15, 25, p. 41-43

Assignment #2--Due Monday, September 28

Stein and Shakarchi: Exercises 5, 16, 22, 28 p. 39-44

Assignment #3--Due Monday, October 25

Stein and Shakarchi: Exercises Chapter 1: 17 and Chapter 2: 1, 9, 10, 11

Assignment #4--Due Wednesday, November 11

Stein and Shakarchi, Exercises in Chapter 2: 4, 7, 17, 19

Assignment #5, Part A--Due Friday, December 11

Stein and Shakarchi, Exercises in Chapter 6: 1, 2, 3, 13

Part B: Also due Friday, December 11

Stein and Shakarchi, Exercise in Chapter 3: 7