# Math 563, Measure Theory, Fall 2016

## General Information

Instructor: Matthew Blair

Email Address: blair ["at"] math.unm.edu

Course Web Page: http://www.math.unm.edu/~blair/math563_f16.html

Office: SMLC 330

Office Hours: Monday 3-5pm and Tuesday 2:15-3:15pm.
**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. L^p spaces and other topics will be covered if time permits.

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

**Meeting times/location:** MWF 1-1:50pm, SMLC 356.

**Prerequisites:** Real analysis at the level of Math 510. 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 Wednesday, September 7

See handout

Assignment #2--Due Wednesday, September 14

See handout

Assignment #3--Due Wednesday, September 28

See handout

Assignment #4--Due Wednesday, October 12

See handout

Assignment #5--Due Wednesday, October 26

See handout

Assignment #6--Due Wednesday, November 9

See handout

Assignment #7--Due Wednesday, November 23

See handout

Assignment #8--Due Friday, December 9

See handout