## AMS Graduate Studies in Mathematics, Sept. 2006.

This text is based on lectures given in the basic graduate measure theory course, Math 203 at UNC. Several goals determined the structure of the notes:

- Quickly get to the construction of Lebesgue measure on the real line
- Then quickly get to the basic results on the Lebesgue integral
- Follow this with further constructions of measures, on Euclidean space, manifolds, etc.
- Then include some more advanced topics

## Contents

- The Riemann integral
- Lebesgue measure on the line
- Integration on measure spaces
- L
^{p}spaces - The Caratheodory construction of measures
- Product measures
- Lebesgue measure on R
^{n}and on manifolds - Signed measures and complex measures
- L
^{p}spaces, II - Sobolev spaces
- Maximal functions and a.e. phenomena
- Hausdorff’s r-dimensional measures
- Radon measures
- Ergodic theory
- Probability spaces and random variables
- Wiener measure and Brownian motion
- Conditional expectation and martingales

**Appendices**

- Metric spaces, topological spaces, and compactness
- Derivatives, diffeomorphisms, and manifolds
- The Whitney extension theorem
- The Marcinkiewicz Interpolation Theorem
- Sard’s theorem
- A change of variable theorem for many-to-one maps
- Integration of differential forms
- Change of variables revisited
- The Gauss-Green formula on Lipschitz domains