Selection Theorems and their Applications
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263 T. Parthasarathy Indian Statistical Institute, Calcutta/India
Selection Theorems and their Applications
Springer-Verlag Berlin· Heidelberg· New York 1972
Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZOrich
263 T. Parthasarathy Indian Statistical Institute, Calcutta/India
Selection Theorems and their Applications
Springer-Verlag Berlin· Heidelberg· New York 1972
AMS Subject Classifications (1970): 49 A 35, 54C65, 90D 15
ISBN 3-540-05818-4 Springer-Verlag Berlin' Heidelberg· New York ISBN 0-387-05818-4 Springer-Verlag New York· Heidelberg· Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin' Heidelberg 1972. Library of Congress Catalog Card Number 72-78192. Printed in Germany. Offsetdruck: Julius Beltz, HemsbachiBergstr.
PREFACE
This volume of lecture notes contains results on selection theorems and their applications. Some of the material of this volume had been given as seminar talks at Case Western Reserve University during 1969 - 1970. This volume contains nine sections each of them followed by selected references. We hope this volume will be profitable to specialists in game theory, dynamic programming, control theory, mathematical economics as well as to all mathematicians interested in this area of Mathematics. I wish to express my sincere thanks to the following Professors: Henry Hermes, Marc Jacobs, Ashok Maitra and Sam Nadler Jr., for several useful suggestions. A particular measure of gratitude is due to Mr. Arun Das who patiently and accurately prepared the final typescript of this volume. My wife Ranjani proof-read the manuscript. To her lowe my heart-felt thanks. Finally I wish to express my gratitude to the Indian Statistical Institute for providing the excellent research facilities and to Springer - Verlag for undertaking the publication of these notes.
T. Parthasarathy November 20, 1971
Indian Statistical Institute
CONTENTS
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Continuous selections. . . . . . . . • • . . . . . . . • . .
2. Continuous selections on metric continuum • • • • •
3. Continuous selections and solutions of generalised differential equations. • . • . . .
. .....
3 14
24
4. Measurable selections and an application to stochastic games.
35
5. General theorems on selectors
49
6. Two applications of measurable selections . • • • • • • • ••
58
7. von-Neumann's measurable choice theorem • • .
65
8. On the uniformization of sets in topological spaces . . . . .
76
9. Supplementary remarks on selection the
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