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基本信息
- 原书名：Real Analysis,Third Edition
- 原出版社： Prentice Hall/Pearson
编辑推荐
在过去的40多年中，本书已被国外众多著名大学(如斯坦福大学、哈佛大学等)采用。本书的题材是数学教学的共同基础，包含许多数学家的研究成果。
内容简介
目录
Prologue to the Student 1
I Set Theory 6
1 Introduction 6
2 Functions 9
3 Unions, intersections, and complements 12
4 Algebras of sets 17
5 The axiom of choice and infinite direct products 19
6 Countable sets 20
7 Relations and equivalences 23
8 Partial orderings and the maximal principle 24
9 Well ordering and the countable ordinals 26
Part One
THEORY OF FUNCTIONS OF A
REAL VARIABLE
2 The Real Number System 31
1 Axioms for the real numbers 31
2 The natural and rational numbers as subsets of R 34
3 The extended real numbers 36
4 Sequences of real numbers 37
5 Open and closed sets of real numbers 40
I Set Theory 6
1 Introduction 6
2 Functions 9
3 Unions, intersections, and complements 12
4 Algebras of sets 17
5 The axiom of choice and infinite direct products 19
6 Countable sets 20
7 Relations and equivalences 23
8 Partial orderings and the maximal principle 24
9 Well ordering and the countable ordinals 26
Part One
THEORY OF FUNCTIONS OF A
REAL VARIABLE
2 The Real Number System 31
1 Axioms for the real numbers 31
2 The natural and rational numbers as subsets of R 34
3 The extended real numbers 36
4 Sequences of real numbers 37
5 Open and closed sets of real numbers 40
前言
In the twenty years since this work was last revised it has contrib-uted to the education of several generadons of students, despite flaws in its treatment of Baire measure and the omission of invariant measures. I am therefore pleased by the opportunity of presenting a new edition ameliorating these shortcomings. Because of the difficulty of recapturing a point of view from the time of original composition,when the material was new to me, I have confined significant
revisions to the theory of locally compact spaces and the study of measures on topological spaces. Elsewhere the alterations consist largely of minor improvements and the addition of new problems.
Part One is almost unchanged, the most notable change being the treatment of the Minkowski and Holder inequalities. This treatment seems to me more natural, and it immediately gives the reversed inequalities for 0 [ p [ 1. There are also relatively few changes in Chapters 11 and 12, the basic chapters on measure and integration.The principal additions consist of a section on integral operators and a section on Hausdorff measure added to Chapter 12.
Part Two sees somewhat more reorganization and extension: The sections on compact metric spaces and the Ascoli theorem have been moved from the chapter on compact spaces to the chapter on metric spaces, making these topics independent of the general theory of topological spaces. The material on Baire Category has been expanded with an indication of the principles used in applying this theory to proofs.
Chapter 8 on topological spaces is virtually unchanged, but Chapter 9 has been largely rewritten to expand the treatment accorded to locally compact spaces. Properties of locally compact spaces needed for measure theory are developed, and the concepts of paracompactness, exhaustion, and a-compactness are discussed at length in the context of locally compact Hausdorff spaces. There is also a section on manifolds and the significance of paracompactness for them.
Chapter 10 is again little changed, with the exception of some material on convexity.
The material on Baire and Borel measures in locally compact spaces has been entirely rewritten. The treatment in the 2nd edition was seriously flawed. I do not think there were any actual misstate ments of fact in the theorems and propositions, but the text was misleading, and a number of the statements in the problems were false. The difficulties, as in some other published treatments of measures in spaces that are not a-compact, arose from problems of regu-
larity. They were caused in my case by a misguided attempt to avoid talking about regularity directly. The current treatment meets these problems face-to-face and shows that one can have Baire (or Borel) measures that are inner regular or that are quasi regular but not always ones that are both. Included with this material is a direct proof of the Riesz-Markoff Theorem on the structure of positive linear functionals on Co(X). This proof is independent of the Daniell
integral, allowing the chapter on the Daniell integral to be relegated to the end of the book.
Chapter 15 on automorphisms of measure spaces has been largely rewritten so that it now gives an extended treatment of Borel measures on complete separable metric spaces. I have tried to please my friend George Mackey by stressing the equivalence of these spaces with certain standard measure spaces, essentially Lebesgue measure on an interval of R.
The present edition contains a new chapter on invariant measures in Part Three. This topic was omitted from earlier editions because I was unsatisfied with the usual development of the theory. I thought the standard presentations of Haar measure awkward in the manner of their use of the Axiom of Choice to assure additivity, and I wanted to use instead a suitable generalization of the notion of limit along the lines used by Banach in the separable metric case. I also
believed the proper context for invariant measures to be that of a transitive group of homeomorphisms on a locally compact space X.Thus the topology should be on the homeogeneous space X, with the group of homeomorphisms an abstract group without topology.Oi course, the group must satisfy some conditions in order that there should be a Baire measure on X invariant under the group. I introduce a property, called topological equicontinuity, and show that it
suffices for the existence of an invariant measure. The unicity of such measures is considered in a number of particular cases, including that of locally compact topological groups. We also consider groups of diffeomorphisms and introduce the Hurwitz invariant integral when it exists. This integral has the advantage that one can give specific formulas for the integrand in many cases.
When this book was originally planned and written, the theory of Lebesgue integration was generally considered to be graduate level material, and the book was designed to be covered in a year-long course for first-year graduate students. Since that time the undergraduate curriculum has tended to include material on Lebesgue integration for advanced students, and this book has found increasing use at this level. The material presented here is of varying difficulty and sophistication. I have tried to arrange the chapters with considerable independence so the book will be useful for a variety of courses. One possibility for a short course is to cover Part One and Chapters 11 and 12. This gives a thorough treatment of integration and differentiation on R together with the fundamentals of abstract measure and integration. This could be supplemented by Chapter 7, covering metric spaces, and some topics on Banach spaces from Chapter 10. For students who are already familiar with basic measure and integration theory as well as the elements of metric spaces, one could construct a short course on measure and integration in topological spaces covering Chapters 9, 13, 14, and 15 with Chapters 7 and 8 as background material.
Were I writing the chapter on set theory today, I would give it a different tone, emphasize the various philosophical points of view about the foundations of mathematics and warn against endowing sets with reality and significance apart from the formal system in which they are embedded. The temptation to rewrite the chapter along these lines has been resisted, but I hope the readers of this book will ultimately read some of the many books on the foundations of mathematics before coming to a fixed opinion on the nature of infinite sets.
I wish to thank all of the diligent readers who have given me corrections and improvements over the last twenty years. My special thanks go to Jay Jorgenson and Hala Khuri for proofreading and checking the work in galley proof and to Elizabeth Arrington and Elizabeth Harvey for turning large amounts of handwritten corrections and material into copy suitable for the printer.
H. L. R.
Stanford, California
July 1987
This book is the outgrowth of a course at Stanford entitled “Theory of Functions of a Real Variable," which I have given from time to time during the last ten years. This course was designed for first-year graduate students in mathematics and statistics. It presup-poses a general background in undergraduate mathematics and specific acquaintance with the material in an undergraduate course on the fundamental concepts of analysis. I have attempted to cover the
revisions to the theory of locally compact spaces and the study of measures on topological spaces. Elsewhere the alterations consist largely of minor improvements and the addition of new problems.
Part One is almost unchanged, the most notable change being the treatment of the Minkowski and Holder inequalities. This treatment seems to me more natural, and it immediately gives the reversed inequalities for 0 [ p [ 1. There are also relatively few changes in Chapters 11 and 12, the basic chapters on measure and integration.The principal additions consist of a section on integral operators and a section on Hausdorff measure added to Chapter 12.
Part Two sees somewhat more reorganization and extension: The sections on compact metric spaces and the Ascoli theorem have been moved from the chapter on compact spaces to the chapter on metric spaces, making these topics independent of the general theory of topological spaces. The material on Baire Category has been expanded with an indication of the principles used in applying this theory to proofs.
Chapter 8 on topological spaces is virtually unchanged, but Chapter 9 has been largely rewritten to expand the treatment accorded to locally compact spaces. Properties of locally compact spaces needed for measure theory are developed, and the concepts of paracompactness, exhaustion, and a-compactness are discussed at length in the context of locally compact Hausdorff spaces. There is also a section on manifolds and the significance of paracompactness for them.
Chapter 10 is again little changed, with the exception of some material on convexity.
The material on Baire and Borel measures in locally compact spaces has been entirely rewritten. The treatment in the 2nd edition was seriously flawed. I do not think there were any actual misstate ments of fact in the theorems and propositions, but the text was misleading, and a number of the statements in the problems were false. The difficulties, as in some other published treatments of measures in spaces that are not a-compact, arose from problems of regu-
larity. They were caused in my case by a misguided attempt to avoid talking about regularity directly. The current treatment meets these problems face-to-face and shows that one can have Baire (or Borel) measures that are inner regular or that are quasi regular but not always ones that are both. Included with this material is a direct proof of the Riesz-Markoff Theorem on the structure of positive linear functionals on Co(X). This proof is independent of the Daniell
integral, allowing the chapter on the Daniell integral to be relegated to the end of the book.
Chapter 15 on automorphisms of measure spaces has been largely rewritten so that it now gives an extended treatment of Borel measures on complete separable metric spaces. I have tried to please my friend George Mackey by stressing the equivalence of these spaces with certain standard measure spaces, essentially Lebesgue measure on an interval of R.
The present edition contains a new chapter on invariant measures in Part Three. This topic was omitted from earlier editions because I was unsatisfied with the usual development of the theory. I thought the standard presentations of Haar measure awkward in the manner of their use of the Axiom of Choice to assure additivity, and I wanted to use instead a suitable generalization of the notion of limit along the lines used by Banach in the separable metric case. I also
believed the proper context for invariant measures to be that of a transitive group of homeomorphisms on a locally compact space X.Thus the topology should be on the homeogeneous space X, with the group of homeomorphisms an abstract group without topology.Oi course, the group must satisfy some conditions in order that there should be a Baire measure on X invariant under the group. I introduce a property, called topological equicontinuity, and show that it
suffices for the existence of an invariant measure. The unicity of such measures is considered in a number of particular cases, including that of locally compact topological groups. We also consider groups of diffeomorphisms and introduce the Hurwitz invariant integral when it exists. This integral has the advantage that one can give specific formulas for the integrand in many cases.
When this book was originally planned and written, the theory of Lebesgue integration was generally considered to be graduate level material, and the book was designed to be covered in a year-long course for first-year graduate students. Since that time the undergraduate curriculum has tended to include material on Lebesgue integration for advanced students, and this book has found increasing use at this level. The material presented here is of varying difficulty and sophistication. I have tried to arrange the chapters with considerable independence so the book will be useful for a variety of courses. One possibility for a short course is to cover Part One and Chapters 11 and 12. This gives a thorough treatment of integration and differentiation on R together with the fundamentals of abstract measure and integration. This could be supplemented by Chapter 7, covering metric spaces, and some topics on Banach spaces from Chapter 10. For students who are already familiar with basic measure and integration theory as well as the elements of metric spaces, one could construct a short course on measure and integration in topological spaces covering Chapters 9, 13, 14, and 15 with Chapters 7 and 8 as background material.
Were I writing the chapter on set theory today, I would give it a different tone, emphasize the various philosophical points of view about the foundations of mathematics and warn against endowing sets with reality and significance apart from the formal system in which they are embedded. The temptation to rewrite the chapter along these lines has been resisted, but I hope the readers of this book will ultimately read some of the many books on the foundations of mathematics before coming to a fixed opinion on the nature of infinite sets.
I wish to thank all of the diligent readers who have given me corrections and improvements over the last twenty years. My special thanks go to Jay Jorgenson and Hala Khuri for proofreading and checking the work in galley proof and to Elizabeth Arrington and Elizabeth Harvey for turning large amounts of handwritten corrections and material into copy suitable for the printer.
H. L. R.
Stanford, California
July 1987
This book is the outgrowth of a course at Stanford entitled “Theory of Functions of a Real Variable," which I have given from time to time during the last ten years. This course was designed for first-year graduate students in mathematics and statistics. It presup-poses a general background in undergraduate mathematics and specific acquaintance with the material in an undergraduate course on the fundamental concepts of analysis. I have attempted to cover the
