Measure and Integral
Jan Maly, Jaroslav Lukes :
Measure and Integral

MatfyzPRESS
(pp. 178, ISBN 80-85863-06-5)

The introductory part, where the abstract theory of measure and integral is discussed, is followed by a chapter on integration in locally compact spaces. This culminates with the Riesz representation theorem. A brief introduction to measure theory on groups (Haar measure) is appended to this chapter. The study on integration on the real line contains the Lebesgue differentiation theorem and the Henstock-Kurzweil integral. From the topics on measure and integration on R^n, the covering theorems, differentiation of measures, density topology and approximately continuous functions are included. New proofs of the Rademacher and Besicovitch theorems are presented. A part of the book is devoted to the study of the theory of distributions, Fourier transform, approximation in function spaces and degree theory. The presentation on the line and surface integral is based on k-dimensional measure (possibly Hausdorff measure) and change of variables for Lipschitz surfaces. The chapter proceeds from an elementary approach (gradient, divergence, rotation) to more advanced one (differential forms on manifolds). The book is finished with integration of Banach space valued functions, where the Bochner, Pettis and Dunford integrals are discussed. Each chapter contains notes and remarks. An ample list of references is included at the end of the book. The text is intended for graduate and senior undergraduate students and young researchers.

CONTENT :

  1. Measures and Measurable Functions
  2. The Abstract Lebesgue Integral
  3. Radon Integral and Measure
  4. Integration on R
  5. Integration on Rn
  6. Change of Variable Formula and k-dimensional Measures
  7. Surface and Curve Integrals
  8. Vector Integration
  9. Appendix on Topology
  10. Bibliography and References
  11. A Short Guide to the Notation
  12. Subject Index