实分析和抽象分析

Chapter One： Set Theory and Algebra

Section 1. The algebra of sets

Section 2. Relations and functions

Section 3. The axiom of choice and ordinal numbers

Section 4. Cardinal numbers and ordinal numbers

Section 5. Construction of the real and complex number fields

Chapter Two： Topology and Continuous Functions

Section 8. The riemann-Stieltjes integral

Section 9. Extending certain functionals

Section 10. Measures and measurable sets

Section 11. Measurable functions

Section 12. The abstract Lebesgue integral

Chapter Four： Function Spaces and Banach Spaces

Section 13. The spaces

Section 14. Abstract Banach spaces

Section 15. The conjugate space

Section 16. Abstract Hilbert spaces

Chapter Five： Differentiation

Section 17. Differentiable and nondifferentiable functions

Section 18. Absolutely continuous functions

Section 19. Complex measures and the LEBESGUE-RADON-NIKODYM theorem

Section 20. Applications of the LEBESGUE-RADON-NIKODYM theorem

Chapter Six： Integration on Product Spaces

Section 21. The product of two measure spaces

Section 22. Products of infinitely many measure spaces

Index of Symbols

Index of Authors and Terms

This book is first of all designed as a text for the course usually called "theory of functions of a real variable". This course is at present customarily offered as a first or second year graduate course in United States universities， although there are signs that this sort of analysis will soon penetrate upper division undergraduate curricula. We have included every topic that we think essential for the training of analysts， and we have also gone down a number of interesting bypaths. We hope too that the book will be useful as a reference for mature mathematicians and other scientific workers. Hence we have presented very general and complete versions of a number of important theorems and constructions. Since these sophisticated versions may be difficult for the beginner， we ave given elementary avatars of all important theorems， with appropriate suggestions for skipping. We have given complete definitions， explanations， and proofs throughout， so that the book should be usable for individual study as well as for a course text