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L 5 Series in Pure Mathematics - Volume 5 MEASURES O N INFINITE DIMENSIONAL SPACES Y Yamasaki World Scientific 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 World ScientiHc Publi^ing Co. Pte. Ltd. P. O. Box 128, Fairer Road, Singapore 9128 242, Cherry Street, Philadelphia PA 19106-1906, USA library of Congress Cataloging in Publication Data Yamasaki, Yasuo, 1934- Lecture 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 photo- copying, 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 PART A: EXTENDABILITY OF A FAMILY OF MEASURES TO A a-ADDITIVE MEASURE (Kolmogorov- Bochner - Minlos Theory ) Introduction..................................................................................................................................................... 3 Chapter 1. Preliminary discussions....................................................................................................... 4 § 1. Explanation of the problem................................................................................................................. 4 §2. Tychonov’s theorem............................................................................................................................. 6 §3. Hopfs theorem............................................................................................................. 9 Chapter 2. Direct product and projective lim it................................................................................. 14 §4. Measurable spaces, their product and limit.......................................................................................... 14 §5. Extension problems and counter examples ........................................................................................ 17 §6. Extension theorem for direct product.................................................................................................. 22 §7. Extension theorem for projective limit............................................................................................... 26 §8. Non-countable direct product and projective limit........................................................................... 30 §9. Compact regular measurable space....................................................................................................... 35 §10. Borel field............................................................................................................................................... 38 §11. Baire field............................................................................................................................... 42 § 12. Product measure..................................................................................................................................... 46 §13. Suslin set, Luzin set .............................................................................................................................. 50 §14. Standard measurable space.................................................................................................................... 56 Chapters. Measures on vector spaces................................................................................................. 63 §15. Explanation of the problem................................................................................................................. 63 § 16. Relation with Bochner’s theorem.......................................................................................................... 68 §17. Minlos’theorem..................................................................................................................................... 72 §18. Sazonov topology................................................................................................................................... 77 §19. Supplementary results........................................................................................................................... 81 §20. Nuclear space.......................................................................................................................................... 85 §21. Heredity.................................................................................................................................................. 89 §22. Dual space............................................................................................................................................... 95 Appendix. Definition of nuclearity without using Hilbertian semi-norms............................. 98 Note.................................................................................................................................................................... 106 PART B: INVARIANCE AND QUASI-INVARIANCE OF MEASURES ON INFINITE DIMENSIONAL SPACES Introduction..................................................................................................................................................... Ill Chapter 1. Invariant measure on a group............................................................................................. 113 § 1. Measurable group, invariant and quasi-invariant measure................................................................. 113 VIII §2. Haar measure on a locally compact group.......................................................................................... 118 §3. Haar measure on a thick group............................................................................................................ 125 §4. Weil topology ...................................................................................................................................... 133 §5. Case of a vector space........................................................................................................................... 138 Chapter 2. Gaussian measures and related problems...................................................................... 144 §6. Quasi-invariance and ergodicity........................................................................................................... 144 §7. Absolute continuity of projective limit measures .............................................................................. 147 §8. Gaussian measures ............................................................................................................................... 151 §9. E'-quasi-invariance and E'-ergodicity.................................................................................................. 154 §10. Mutual equivalence............................................................................................................................... 160 §11. Rotationally invariant measures........................................................................................................... 163 §12. Representation of L^(jLi)..................................................................................................................... 167 Chapter 3. The set of all quasi-invariant translations.............................. ............................ 174 §13. Convolution of measures..................................................................................................................... 174 § 14. Linearization of a topology on a vector space..................................................................................... 175 §15. Characteristic topology........................................................................................................................ 177 § 16. Evaluation of in terms of ........................................................................................................... 181 § 17. Some applications.................................................................................................................................. 184 §18. Kakutani topology............................................................................................................................... 189 § 19. Evaluation of Y^ in terms of Kakutani topology................................................................................ 195 Chapter 4. Product measures on IR” ....................................................................... 202 §20. Product of one-dimensional probability measures............................................................................. 202 §21. Stationary product measures................................................................................................................. 208 §22. Gaussian measures and stationary products....................................................................................... 212 § 23. Estimation of Y^^ and R^^...................................................................................................................... 216 §24. Non-stationary product measures ....................................................................................................... 219 §25. (£^)-quasi-invariance........................................................................................................................... 222 Chapter 5. IR“ -invariant measures on R“ ........................................................................................... 227 §26. Infinite dimensional Lebesgue measure............................................................................................... 227 §27. IRÔ-ergodicity and mutual equivalence............................................................................................... 230 §28. Equivalent probability measure of product type................................................................................ 233 §29. The converse problem of §28 .............................................................................................................. 235 §30. Linear transformation of ................................................................................................................. 240 §31. Rotational invariance and ergodicity .................................................................................................. 244 §32. Invariance under homotheties............................................................................................................... 247 Note.................................................................................................................................................................. 251 Index................................................................................................................................................................ 255