Measure Theory and Integration

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

  1. The Riemann integral
  2. Lebesgue measure on the line
  3. Integration on measure spaces
  4. Lp spaces
  5. The Caratheodory construction of measures
  6. Product measures
  7. Lebesgue measure on Rn and on manifolds
  8. Signed measures and complex measures
  9. Lp spaces, II
  10. Sobolev spaces
  11. Maximal functions and a.e. phenomena
  12. Hausdorff’s r-dimensional measures
  13. Radon measures
  14. Ergodic theory
  15. Probability spaces and random variables
  16. Wiener measure and Brownian motion
  17. 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