Real analysis.
4th ed.

*(English)*Zbl 1191.26002
International Edition. New York, NY: Prentice Hall (ISBN 978-0-13-511355-4/pbk). xii, 505 p. (2010).

This is the fourth edition [for the first editions see I: New York: The Macmillan Company; London: Collier-Macmillan, Ltd. (1963; Zbl 0121.05501); II: London: The Macmillan Company XII (1969; Zbl 0197.03501); III: New York: Macmillan Publishing Company; London: Collier Macmillan Publishing (1988; Zbl 0704.26006)] of the well-known treatise in Real Analysis written by H. Royden. This edition is essentially enlarged and improved by Patrick M. Fitzpatrick. Really, this is a new book, although its aim is the same: to present the measure theory, integration theory, and functional analysis that a modern analyst needs to know. The structure of this is also the same: Part I treats Lebesgue measure and Lebesgue integration for functions of a single variable; Part II treats abstract spaces – topological, metric, Banach, Hilbert spaces; Part III treats integration over general measure spaces, together with the enhancement that the general theory possesses in the presence of a topological, algebraic, or dynamical structure. Below the list of basic changes in this edition is presented:

Part I – the concept of uniform integrability and the Vitali convergence theorem, briefly Cauchy sequences in spaces \(L^p(E)\), weak compactness in \(L^p(E)\) with applications to extremum problems for continuous convex functionals; Part II – general structural properties and corresponding principal theorems of metric and topological spaces are now presented separately, the duality between a Banach space and its dual and corresponding properties of the weak topology are systematized, a new chapter about Hilbert spaces, operators acting in them, and linear operator equations with such operators; Part III – general measure theory and general integration theory in general measure spaces are developed including the Lebesgue spaces and representation of their duals, the basic properties of weak topologies in Lebesgue spaces, Dunford-Pettis theorem and some others, the description of the dual of \(C(X)\) based on the analysis of the relationship between topology and measure on \(X\), for a compact Hausdorff space \(X\). Other changes in the book in general: some fundamental results as Egoroff’s theorem and Urysohn lemma are now proven in the text, Borel-Cantelli’s lemma, Chebyshev’s inequality, Luzin’s theorem and other results about continuity properties of measure and integral are now presented in the text along with several other concepts.

Here is Contents of this edition. Part I, “Lebesgue integration for functions of a single real variable”: Preliminaries on sets, mappings, and relations, 1. The real numbers: sets, sequences and functions, 2. Lebesgue measure, 3. Lebesgue measurable functions, 4. Lebesgue integration, 5. Lebesgue integration: further topics, 6. Differentiation and integration, 7. The \(L^p\) spaces: completeness and approximation 8. The \(L^p\) spaces: duality and weak compactness.

Part II, “Abstract spaces: metric, topological, Banach, and Hilbert spaces”: 9. Metric spaces: general properties, 10. Metric spaces: three fundamental theorems, 11. Topological spaces: general properties, 12. Topological spaces: three fundamental theorems, 13. Continuous linear operators between Banach spaces, 14. Duality for normed linear spaces, 15. Compactness results: the weak topology, 16. Continuous linear operators on Hilbert spaces.

Part III, “Measure and integration: general theory”: 17. General measure spaces: their properties and construction, 18. Integration over general measure spaces, 19. General \(L^p\) spaces: completeness, duality, and weak convergence, 20. The construction of particular measures, 21. Measure and topology, 22. Invariant measures. Bibliography. Index.

I think that this edition is interesting and will be useful for all specialists in Analysis: lecturers, graduate and post graduate students, and also all mathematicians who deal with analysis and its applications. Doubtlessly, this book will also be useful in any serious mathematical library.

Part I – the concept of uniform integrability and the Vitali convergence theorem, briefly Cauchy sequences in spaces \(L^p(E)\), weak compactness in \(L^p(E)\) with applications to extremum problems for continuous convex functionals; Part II – general structural properties and corresponding principal theorems of metric and topological spaces are now presented separately, the duality between a Banach space and its dual and corresponding properties of the weak topology are systematized, a new chapter about Hilbert spaces, operators acting in them, and linear operator equations with such operators; Part III – general measure theory and general integration theory in general measure spaces are developed including the Lebesgue spaces and representation of their duals, the basic properties of weak topologies in Lebesgue spaces, Dunford-Pettis theorem and some others, the description of the dual of \(C(X)\) based on the analysis of the relationship between topology and measure on \(X\), for a compact Hausdorff space \(X\). Other changes in the book in general: some fundamental results as Egoroff’s theorem and Urysohn lemma are now proven in the text, Borel-Cantelli’s lemma, Chebyshev’s inequality, Luzin’s theorem and other results about continuity properties of measure and integral are now presented in the text along with several other concepts.

Here is Contents of this edition. Part I, “Lebesgue integration for functions of a single real variable”: Preliminaries on sets, mappings, and relations, 1. The real numbers: sets, sequences and functions, 2. Lebesgue measure, 3. Lebesgue measurable functions, 4. Lebesgue integration, 5. Lebesgue integration: further topics, 6. Differentiation and integration, 7. The \(L^p\) spaces: completeness and approximation 8. The \(L^p\) spaces: duality and weak compactness.

Part II, “Abstract spaces: metric, topological, Banach, and Hilbert spaces”: 9. Metric spaces: general properties, 10. Metric spaces: three fundamental theorems, 11. Topological spaces: general properties, 12. Topological spaces: three fundamental theorems, 13. Continuous linear operators between Banach spaces, 14. Duality for normed linear spaces, 15. Compactness results: the weak topology, 16. Continuous linear operators on Hilbert spaces.

Part III, “Measure and integration: general theory”: 17. General measure spaces: their properties and construction, 18. Integration over general measure spaces, 19. General \(L^p\) spaces: completeness, duality, and weak convergence, 20. The construction of particular measures, 21. Measure and topology, 22. Invariant measures. Bibliography. Index.

I think that this edition is interesting and will be useful for all specialists in Analysis: lecturers, graduate and post graduate students, and also all mathematicians who deal with analysis and its applications. Doubtlessly, this book will also be useful in any serious mathematical library.

Reviewer: Peter Zabreiko (Minsk)

##### MSC:

26-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to real functions |

28-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to measure and integration |

46-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis |

47-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory |