Measures On Infinite Dimensional Spaces

L Series in Pure Mathematics - Volume 5 MEASURES O N INFINITE DIMENSIONAL SPACES Y Yam asaki World Scientific 5 MEASURES ON INFINITE DIMENSIONAL SPACES SERIES IN PURE MATHEMATICS Editor: C C Hsiung Associate Editors: S S Chern, S Kobayashi, I Satake, Y-T Siu, W-T Wu and M Yamaguti. P^rt I. Monographs and Textbooks Volume 1: Total Mean Curvature and Submanifolds of Finite Type B YChen Volumes: Structures on Manifolds K Yarn &MKon Volume 4: Goldback Conjecture Wang Yuan ( editor) Part II. Lecture Notes Volume 2: A Survey of Trace Forms of Algebraic Number Fields P E Conner & R Perils Volume 5: Measures on Infinite Dimensional Spaces Y Yamasaki Series in Pure Mathematics— Volume 5 MEASURES ON INFINITE DIMENSIONAL SPACES Y Yamasaki Research Institute for Mathematical Sciences Kyoto University World Scientific Singapore • Philadelphia Published by W orld ScientiH c P ubli^ ing C o. P te. Ltd. P. O. Box 128, Fairer Road, Singapore 9128 242, Cherry Street, Philadelphia PA 19106-1906, USA lib rary o f Congress Cataloging in Publication Data Yamasaki, Yasuo, 1934Lecture Notes on Measures on Infinite Dimensional Spaces. (Series in pure mathematics; v. 5) “Dedicated to Professor Hisaaki Yoshizawa on his 60th birthday and to Professor Shizuo Kakutani on his 70th birthday.” 1. Measure theory. 2. Spaces, Generalized. I. Title. II. Series. QA312.Y36 1985 515.4’2 85-9381 ISBN 9971-978-52-0 Copyright © 1985 by World Scientific Publishing Co Pte Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. Printed in Singapore by Singapore National Printers (Pte) Ltd. INTRODUCTORY NOTE These notes are based on the lectures given at Yale (1979-81) and at Kyoto (1981-82). The author wishes to dedicate these lecture notes to Professor Hisaaki Yoshizawa (Kyoto University) and Professor Shizuo Kakutani (Yale University). The author started his study of functional analysis under the direction of Professor Yoshizawa, who also kindly suggested the author’s visit to Yale. Professor Kakutani arranged so heartily the author’s stay and study at Yale, both stimulating and comfortable for the author. CONTENTS PA R T A: E X T EN D A B IL ITY O F A FA M ILY O F M EA SU RES TO A a -A D D IT IV E M EASURE ( K olm ogorov- B ochner - M inlos T h eo ry ) In tr o d u c tio n ..................................................................................................................................................... 3 C hapter 1. Prelim inary d is c u s s io n s ....................................................................................................... § 1. Explanation of the p ro b lem ................................................................................................................. §2. Tychonov’s th e o re m ............................................................................................................................. §3. Hopfs th e o re m .