Math 535 (Real Analysis). Fall 2014

Instructor: Steve Krone

Office: 421 Brink Hall; 885-6317
Office Hours: by appointment.

Class Time: MWF 10:30 - 11:20
Place: Ren 129

Prerequisite: Advanced Calculus

Text: R. Wheeden and A. Zygmund, Measure and Integral, Dekker

This is the standard graduate course in measure theory and integration. In addition to being important in it's own right and for a broader study of analysis, this subject is essential to any serious study of probability, differential equations, functional analysis, ergodic theory, and fractals.

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

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, Lp spaces and convergence in Lp, Holder's inequality, Fourier transforms, convergence of integrals.
  5. Product measures and Fubini's theorem.
  6. Differentiation of integrals and measures, Radon-Nikodym theorem.
  7. Other topics from real analysis.

Some other texts on real analysis: