This 1965 book is a very thorough text covering many classical topics in analysis. The book’s purpose is summarized on p. 53 as “The main goal of this text is to give a complete presentation of integration and differentiation,” and the book never strays very far from this goal.
The level of abstraction referred to in the second half of the title is not very high: the book deals with general measure spaces but is otherwise not very abstract. It has a moderate amount of material on function spaces, but these are presented to aid the work on integration and differentiation and not as subjects in themselves. For example, the book uses the Daniell approach to integration (through functionals) rather than defining integrals directly in terms of measures. Scattered throughout the book is a lot of material on Fourier analysis, although only on the real line.
The book has an excellent index and is more of a reference than a text. There are a lot of exercises, but some areas get many more than others and they seem to be included more to develop additional topics than to develop the reader's skill. The sequence of theorems is very well organized, so that no individual proof is very long but also every theorem is useful or interesting.
A good book that covers generally the same topics but in much less detail is Boas's A Primer of Real Functions. Another good book that also covers generally the same topics but is more abstract (and perhaps more modern) is Gert K. Pedersen’s Analysis Now (Springer, 1996).
Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at MathNerds.com, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.
Chapter One: Set Theory and Algebra
Section 1. The algebra of sets
Section 2. Relations and functions
Section 3. The axiom of choice and some equivalents
Section 4. Cardinal numbers and ordinal numbers
Section 5. Construction of the real and complex number fields
Chapter Two: Topology and Continuous Functions
Section 6. Topological preliminaries
Section 7. Spaces of continuous functions
Chapter Three: The Lebesgue Integral
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 Lp (1 ≤ p < ∞)
Section 14. Abstract Banach spaces
Section 15. The conjugate space of Lp (1 < p < ∞)
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