## Math 535 (Real Analysis). Fall 2016

**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:** 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%).

**Rough Course Outline:**

- Introduction and motivation, including Lebesgue's idea for constructing
integrals of highly discontinuous functions.

- Measurable sets, sigma-fields, measures and outer measures,
construction of Lebesgue measure on R
^{n}.

- Measurable functions, almost everywhere convergence and convergence
in measure.

- 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.

- L
^{p} spaces, convergence in L^{p}, dual spaces,
Holder's inequality, convolution.

- Fourier transforms.

- Distributions, weak derivatives and weak solutions of differential
equations.

- Other topics from real analysis and applications.

**Some other texts on real analysis:**

- A. Friedman, Foundations of Modern Analysis
- R. Wheeden and A. Zygmund, Measure and Integral
- H.L. Royden, Real Analysis
- W. Rudin, Real and Complex Analysis
- Bruckner, Bruckner and Thomson, Real Analysis