Instructor: Matthew Blair

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

Course Web Page: http://www.math.unm.edu/~blair/math563_f12.html

Office: SMLC 330

Office Hours: TBD

**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:** Tuesdays and Thursdays, 11am-12:15pm, SMLC 352.

**Prerequisites:** Real analysis at the level of Math 510 or possibly 401/501. Consult the instructor with any questions.

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.

HomeworkAssignment #1--Due Thursday, September 13

Stein and Shakarchi: Chapter 1 Exercises 4(a,b), 5, 14, 26, 28, p. 38-44

Assignment #2--Due Thursday, September 27

Stein and Shakarchi: Chapter 1 Exercises 7, 8, 10, 22, p. 39-43

Assignment #3--Due Tuesday, October 16

Stein and Shakarchi: Chapter 2 Exercises 4, 6, 10, 11, 16, 19 p. 90-93

Assignment #4--Due Thursday, November 1

See handout.

Assignment #5--Due Tuesday, November 27

Stein and Shakarchi: Chapter 6 Exercises 2, 3, 13 and do exercise number 1 on your own.