Math 535 (Real Analysis). Fall 2022

Instructor: Steve Krone

Office: 421 Brink
Office Hours: by appointment.

Class Time: MWF 11:30 - 12:20
Place: 315 McClure

Prerequisite: Advanced Calculus

Text: E. Lieb and M. Loss, Analysis, 2nd ed., AMS

This is a graduate course in measure theory and integration, together with important topics in analysis. This subject is essential to any serious study of probability, differential equations, functional analysis, ergodic theory, and fractals. In addition to the development of a "grown-up" theory of integration, the book by Lieb and Loss provides a wonderful account of how analysis is actually used in practice. It goes beyond the usual development of mathematical theory for its own sake and let's you see the theory in action.

Grades will be based on homework (30%), a midterm exam (30%), and a final exam (40%).

Midterm Exam: Fri, Oct 7

Final Exam: Mon, Dec 12, 10:15-12:15

Rough Course Outline:

  1. Introduction and motivation, including Lebesgue's idea for constructing integrals of highly discontinuous functions.
  2. Measurable sets, sigma-fields, measures and outer measures, construction of Lebesgue measure on Rn.
  3. Measurable functions, almost everywhere convergence and convergence in measure.
  4. Lebesgue integral of a function with respect to a measure, properties of integrals, comparison of Riemann and Lebesgue integrals, product measures and Fubini's theorem, convergence of integrals.
  5. Lp spaces, convergence in Lp, dual spaces, Holder's inequality, convolution.
  6. Fourier transforms.
  7. Distributions, weak derivatives and weak solutions of differential equations.
  8. Other topics from real analysis and applications.

Some other texts on real analysis: