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基本信息
- 原书名：Real Analysis and Probability
- 原出版社： Cambridge University Press
编辑推荐
这是一本广受称赞的教科书，清晰地讲解了现代概率论以及度量空间与概率测度之间的相互作用。本书在两个方面获得了极佳的成功。一是它是一本全面、新颖的实分析教程，二是它是一本数学理论完整和自成体系的概率论教程。
内容简介
书籍
数学书籍
本书在两个方面获得了极佳的成功。一是它是一本全面、新颖的实分析教程，二是它是一本数学理论完整和自成体系的概率论教程。本书无疑给出了一种严谨和完整的新标准。.
这是一本非凡的著作。在教学和参考两个方面本书将成为一本标准化教材，它全面地介绍了实分析的必备知识，且证明贯穿全书。一些主题和证明极少在其他教科书中见到。
严谨，精深，新颖，这是一本适用于数学专业研究生的教材。..
这是一本广受称赞的教科书，清晰地讲解了现代概率论以及度量空间与概率测度之间的相互作用。本书分两部分，第一部分介绍了实分析的内容，包括基本集合论、一般拓扑学、测度论、积分法、巴拿赫空间和拓扑空间中的泛函分析导论、凸集和函数、拓扑空间上的测度等。第二部分介绍了基于测度论的概率方面的内容，包括大数律、遍历定理、中心极限定理、条件期望、鞅收敛等。另外，随机过程一章 (第12章) 还介绍了布朗运动和布朗桥。 与前版相比，本版内容更完善，一开始就介绍了实数系的基础和泛代数中的一致逼近的斯通-魏尔斯特拉斯定理；修订和改进了几节的内容，扩充了大量历史注记；增加了很多新的习题，以及对一些习题的解答的提示。...
数学书籍
本书在两个方面获得了极佳的成功。一是它是一本全面、新颖的实分析教程，二是它是一本数学理论完整和自成体系的概率论教程。本书无疑给出了一种严谨和完整的新标准。.
这是一本非凡的著作。在教学和参考两个方面本书将成为一本标准化教材，它全面地介绍了实分析的必备知识，且证明贯穿全书。一些主题和证明极少在其他教科书中见到。
严谨，精深，新颖，这是一本适用于数学专业研究生的教材。..
这是一本广受称赞的教科书，清晰地讲解了现代概率论以及度量空间与概率测度之间的相互作用。本书分两部分，第一部分介绍了实分析的内容，包括基本集合论、一般拓扑学、测度论、积分法、巴拿赫空间和拓扑空间中的泛函分析导论、凸集和函数、拓扑空间上的测度等。第二部分介绍了基于测度论的概率方面的内容，包括大数律、遍历定理、中心极限定理、条件期望、鞅收敛等。另外，随机过程一章 (第12章) 还介绍了布朗运动和布朗桥。 与前版相比，本版内容更完善，一开始就介绍了实数系的基础和泛代数中的一致逼近的斯通-魏尔斯特拉斯定理；修订和改进了几节的内容，扩充了大量历史注记；增加了很多新的习题，以及对一些习题的解答的提示。...
作译者
R. M. Dudley，麻省理工学院数学教授。除本书外，他还著有《Differentiability of Six Operators on Nonsmooth Functions and p-Variation》、《Uniform Central Limit Theorems》等书。...
目录
Preface to the Cambridge Edition .
1 Foundations; Set Theory
1.1 Definitions for Set Theory and the Real Number Systen
1.2 Relations and Orderings
* 1.3 Transfinite Induction and Recursion
1.4 Cardinality
1.5 The Axiom of Choice and Its Equivalents
2 General Topology
2.1 Topologies, Metrics, and Continuity
2.2 Compactness and Product Topologies
2.3 Complete and Compact Metric Spaces
2.4 Some Metrics for Function Spaces
2.5 Completion and Completeness of Metric Spaces
*2.6 Extension of Continuous Functions
*2.7 Uniformities and Uniform Spaces
*2.8 Compactification
3 Measures
3.1 Introduction to Measures
3.2 Semirings and Rings
3.3 Completion of Measures
1 Foundations; Set Theory
1.1 Definitions for Set Theory and the Real Number Systen
1.2 Relations and Orderings
* 1.3 Transfinite Induction and Recursion
1.4 Cardinality
1.5 The Axiom of Choice and Its Equivalents
2 General Topology
2.1 Topologies, Metrics, and Continuity
2.2 Compactness and Product Topologies
2.3 Complete and Compact Metric Spaces
2.4 Some Metrics for Function Spaces
2.5 Completion and Completeness of Metric Spaces
*2.6 Extension of Continuous Functions
*2.7 Uniformities and Uniform Spaces
*2.8 Compactification
3 Measures
3.1 Introduction to Measures
3.2 Semirings and Rings
3.3 Completion of Measures
前言
This is a text at the beginning graduate level. Some study of intermediate analysis in Euclidean spaces will provide helpful background, but in this edition such background is not a formal prerequisite. Efforts to make the book more self-contained include inserting material on the real number system into Chapter 1, adding a treatment of the Stone-Weierstrass theorem, and generally eliminating references for proofs to other books except at very few points, such as some complex variable theory in Appendix B. .
Chapters 1 through 5 provide a one-semester course in real analysis. Following that, a one-semester course on probability can be based on Chapters 8 through 10 and parts of 11 and 12. Starred paragraphs and sections, such as those found in Chapter 6 and most of Chapter 7, are called on rarely, if at all, later in the book. They can be skipped, at least on first reading, or until needed.
Relatively few proofs of less vital facts have been left to the reader. I would be very glad to know of any substantial unintentional gaps or errors. Although I have worked and checked all the problems and hints, experience suggests that mistakes in problems, and hints that may mislead, are less obvious than errors in the text. So take hints with a grain of salt and perhaps make a first try at the problems without using the hints.
I looked for the best and shortest available proofs for the theorems. Short proofs that have appeared in journal articles, but in few if any other textbooks, are given for the completion of metric spaces, the strong law of large numbers, the ergodic theorem, the martingale convergence theorem, the subadditive ergodic theorem, and the Hartman-Wintner law of the iterated logarithm. ..
Around 1950, when Halmos? classic Measure Theory appeared, the more advanced parts of the subject headed toward measures on locally compact spaces, as in, for example, §7.3 of this book. Since then, much of the research in probability theory has moved more in the direction of metric spaces. Chapter 11 gives some facts connecting metrics and probabilities which follow the newer trend. Appendix E indicates what can go wrong with measures
on (locally) compact nonmetric spaces. These parts of the book may well not be reached in a typical one-year course but provide some distinctive material for present and future researchers.
Problems appear at the end of each section, generally increasing in difficulty as they go along. I have supplied hints to the solution of many of the problems. There are a lot of new or, I hope, improved hints in this edition.
I have also tried to trace back the history of the theorems to give credit where it is due. Historical notes and references, sometimes rather extensive, are given at the end of each chapter. Many of the notes have been augmented in this edition and some have been corrected. I don't claim, however, to give the last word on any part of the history.
The book evolved from courses given at M.I.T since 1967 and in Aarhus, Denmark, in 1976. For valuable comments Iam glad to thank Ken Alexander, Deborah Allinger, Laura Clemens, Ken Davidson, Don Davis, Persi Diaconis, Amout Eikeboom, Sy Friedman, David Gillman, Jose Gonzalez, E. Griffor, Leonid Grinblat, Dominique Haughton, J. Hoffmann-Jφrgensen, Arthur Mattuck, Jim Munkres, R. Proctor, Nick Reingold, Rae Shortt, Dorothy Maharam Stone, Evangelos Tabakis, Jin-Gen Yang, and other students and colleagues.
For helpful comments on the first edition I am thankful to Ken Brown, Justin Corvino, Charles Goldie, Charles Hadlock, Michael Jansson, Suman Majumdar, Rimas Norvaisa, Mark Pinsky, Andrew Rosalsky, the late Rae Shoat, and Dewey Tucker. I especially thank Andries Lenstra and Valentin Petrov for longer lists of suggestions. Major revisions have been made to §10.2 (regular conditional probabilities) and in Chapter 12 with regard to Markov times. ...
R. M. Dudley
Chapters 1 through 5 provide a one-semester course in real analysis. Following that, a one-semester course on probability can be based on Chapters 8 through 10 and parts of 11 and 12. Starred paragraphs and sections, such as those found in Chapter 6 and most of Chapter 7, are called on rarely, if at all, later in the book. They can be skipped, at least on first reading, or until needed.
Relatively few proofs of less vital facts have been left to the reader. I would be very glad to know of any substantial unintentional gaps or errors. Although I have worked and checked all the problems and hints, experience suggests that mistakes in problems, and hints that may mislead, are less obvious than errors in the text. So take hints with a grain of salt and perhaps make a first try at the problems without using the hints.
I looked for the best and shortest available proofs for the theorems. Short proofs that have appeared in journal articles, but in few if any other textbooks, are given for the completion of metric spaces, the strong law of large numbers, the ergodic theorem, the martingale convergence theorem, the subadditive ergodic theorem, and the Hartman-Wintner law of the iterated logarithm. ..
Around 1950, when Halmos? classic Measure Theory appeared, the more advanced parts of the subject headed toward measures on locally compact spaces, as in, for example, §7.3 of this book. Since then, much of the research in probability theory has moved more in the direction of metric spaces. Chapter 11 gives some facts connecting metrics and probabilities which follow the newer trend. Appendix E indicates what can go wrong with measures
on (locally) compact nonmetric spaces. These parts of the book may well not be reached in a typical one-year course but provide some distinctive material for present and future researchers.
Problems appear at the end of each section, generally increasing in difficulty as they go along. I have supplied hints to the solution of many of the problems. There are a lot of new or, I hope, improved hints in this edition.
I have also tried to trace back the history of the theorems to give credit where it is due. Historical notes and references, sometimes rather extensive, are given at the end of each chapter. Many of the notes have been augmented in this edition and some have been corrected. I don't claim, however, to give the last word on any part of the history.
The book evolved from courses given at M.I.T since 1967 and in Aarhus, Denmark, in 1976. For valuable comments Iam glad to thank Ken Alexander, Deborah Allinger, Laura Clemens, Ken Davidson, Don Davis, Persi Diaconis, Amout Eikeboom, Sy Friedman, David Gillman, Jose Gonzalez, E. Griffor, Leonid Grinblat, Dominique Haughton, J. Hoffmann-Jφrgensen, Arthur Mattuck, Jim Munkres, R. Proctor, Nick Reingold, Rae Shortt, Dorothy Maharam Stone, Evangelos Tabakis, Jin-Gen Yang, and other students and colleagues.
For helpful comments on the first edition I am thankful to Ken Brown, Justin Corvino, Charles Goldie, Charles Hadlock, Michael Jansson, Suman Majumdar, Rimas Norvaisa, Mark Pinsky, Andrew Rosalsky, the late Rae Shoat, and Dewey Tucker. I especially thank Andries Lenstra and Valentin Petrov for longer lists of suggestions. Major revisions have been made to §10.2 (regular conditional probabilities) and in Chapter 12 with regard to Markov times. ...
R. M. Dudley
