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