Theory of correspondences. Including applications to mathematical economics.

*(English)*Zbl 0556.28012
Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-Interscience Publication. New York etc.: John Wiley & Sons. XI, 256 p. £40.85 (1984).

In recent years correspondences or set-valued maps have vecome a common tool in mathematical economics, control theory, optimization, mathematical programming, and non-smooth analysis, among other areas. This book is the first to bring together a large portion of the basic material about correspondences which is scattered throughout the literature.

The book begins at an elementary level, devoting the first of four parts to the topological background required. The second part is devoted to continuous correspondences, or rather to the various continuity properties that a correspondence may enjoy. The third part deals with measurable correspondences and measurable selections, and the fourth with the integration theory of correspondences. The last chapter in each part gives applications of that part to mathematical economics. This subdivision of the material is very natural and the applications well devised.

Overall the book provides a very detailed development of the foundations of the theory of continuous and measurable correspondences. Each of the twenty chapters ends with a section of notes and remarks which refers the reader to further material in the literature. Those who use the theory of correspondences in their work will find this book a valuable source of information and references. Those who wish to learn about the subject are well advised to take advantage of the enormous amount of time and effort that these authors have put into making this a very readable and well organized presentation of the theory.

There is a good deal of material about correspondences which is not mentioned in this book, e.g. differential inclusions, maximal monotone operators, parameterizations of correspondences, etc. Rather than delve into the numerous applications and more sophisticated aspects of correspondences the book concentrates on presenting the foundational material with indicates of its value in mathematical economics, and this it accomplishes very well.

The book begins at an elementary level, devoting the first of four parts to the topological background required. The second part is devoted to continuous correspondences, or rather to the various continuity properties that a correspondence may enjoy. The third part deals with measurable correspondences and measurable selections, and the fourth with the integration theory of correspondences. The last chapter in each part gives applications of that part to mathematical economics. This subdivision of the material is very natural and the applications well devised.

Overall the book provides a very detailed development of the foundations of the theory of continuous and measurable correspondences. Each of the twenty chapters ends with a section of notes and remarks which refers the reader to further material in the literature. Those who use the theory of correspondences in their work will find this book a valuable source of information and references. Those who wish to learn about the subject are well advised to take advantage of the enormous amount of time and effort that these authors have put into making this a very readable and well organized presentation of the theory.

There is a good deal of material about correspondences which is not mentioned in this book, e.g. differential inclusions, maximal monotone operators, parameterizations of correspondences, etc. Rather than delve into the numerous applications and more sophisticated aspects of correspondences the book concentrates on presenting the foundational material with indicates of its value in mathematical economics, and this it accomplishes very well.

Reviewer: J.E.Jayne

##### MSC:

28B20 | Set-valued set functions and measures; integration of set-valued functions; measurable selections |

28-02 | Research exposition (monographs, survey articles) pertaining to measure and integration |

54-02 | Research exposition (monographs, survey articles) pertaining to general topology |

54C60 | Set-valued maps in general topology |

54C65 | Selections in general topology |

91B38 | Production theory, theory of the firm |

91B50 | General equilibrium theory |