Math 535 (Real Analysis). Fall 2014
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,
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%).
- 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 Rn.
- Measurable functions, almost everywhere convergence and convergence
- 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.
- 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