Measures On Infinite Dimensional Spaces

270 Pages·1985·6.441 MB·English

by  Yamasaki, Y.

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