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
   • The Lebesgue Measure
   • Abstract Measures
   • Measurable Functions
   • Construction of Measures from Outer Measures
   • Classes of Sets and Set Functions
   • Signed and Complex Measures
2. The Abstract Lebesgue Integral
   • Integration on R
   • The Abstract Lebesgue Integral
   • Integrals Depending on a Parameter
   • The L^p Spaces
   • Product Measures and the Fubini Theorem
   • Sequences of Measurable Functions
   • The Radon-Nikodym Theorem and the Lebesgue Decomposition
3. Radon Integral and Measure
   • Radon Integral and Radon Outer Measure
   • Radon Measures
   • Riesz Representation Theorem
   • Sequences of Measures
   • Luzin's Theorem
   • Measures on Topological Groups
4. Integration on R
   • Integral and Differentiation
   • Functions of Finite Variation and Absolutely Continuous Functions
   • Theorems on Almost Everywhere Differentiation
   • Indefinite Lebesgue Integral and Absolute Continuity
   • Radon Measures on R and Distribution Functions
   • Henstock-Kurzweil Integral
5. Integration on Rn
   • Lebesgue Measure and Integral on Rn
   • Covering Theorems
   • Differentiation of Measures
   • Lebesgue Density Theorem and Approximately Continuous Functions
   • Lipschitz Functions
   • Approximation Theorems
   • Distributions
   • Fourier Transform
6. Change of Variable Formula and k-dimensional Measures
   • Change of Variable Theorem
   • The Degree of a Mapping
   • Hausdorff Measures
7. Surface and Curve Integrals
   • Integral Calculus in Vector Analysis
   • Integration of Differential Forms
   • Integration on Manifolds
8. Vector Integration
   • Measurable functions
   • Vector Measures
   • The Bochner Integral
   • The Dunford and Pettis Integrals
9. Appendix on Topology
10. Bibliography and References
11. A Short Guide to the Notation
12. Subject Index