## 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:**

- 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,
L
^{p} spaces and convergence in L^{p}, Holder's inequality,
Fourier transforms, convergence of integrals.

- Product measures and Fubini's theorem.

- Differentiation of integrals and measures, Radon-Nikodym theorem.

- Other topics from real analysis.

**Some other texts on real analysis:**

- A. Friedman, Foundations of Modern Analysis
- H.L. Royden, Real Analysis
- W. Rudin, Real and Complex Analysis
- Bruckner, Bruckner and Thomson, Real Analysis