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Topics in the Theory of Lifting

  • A. Ionescu Tulcea
  • C. Ionescu Tulcea

Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE2, volume 48)

Table of contents

  1. Front Matter
    Pages I-X
  2. A. Ionescu Tulcea, C. Ionescu Tulcea
    Pages 1-19
  3. A. Ionescu Tulcea, C. Ionescu Tulcea
    Pages 20-33
  4. A. Ionescu Tulcea, C. Ionescu Tulcea
    Pages 34-42
  5. A. Ionescu Tulcea, C. Ionescu Tulcea
    Pages 43-53
  6. A. Ionescu Tulcea, C. Ionescu Tulcea
    Pages 54-66
  7. A. Ionescu Tulcea, C. Ionescu Tulcea
    Pages 67-85
  8. A. Ionescu Tulcea, C. Ionescu Tulcea
    Pages 86-103
  9. A. Ionescu Tulcea, C. Ionescu Tulcea
    Pages 104-132
  10. A. Ionescu Tulcea, C. Ionescu Tulcea
    Pages 133-154
  11. A. Ionescu Tulcea, C. Ionescu Tulcea
    Pages 155-170
  12. Back Matter
    Pages 171-192

About this book

Introduction

The problem as to whether or not there exists a lifting of the M't/. 1 space ) corresponding to the real line and Lebesgue measure on it was first raised by A. Haar. It was solved in a paper published in 1931 [102] by 1. von Neumann, who established the existence of a lifting in this case. In subsequent papers J. von Neumann and M. H. Stone [105], and later on 1. Dieudonne [22], discussed various algebraic aspects and generalizations of the problem. Attemps to solve the problem as to whether or not there exists a lifting for an arbitrary M't/. space were unsuccessful for a long time, although the problem had significant connections with other branches of mathematics. Finally, in a paper published in 1958 [88], D. Maharam established, by a delicate argument, that a lifting of M't/. always exists (for an arbi­ trary space of a-finite mass). D. Maharam proved first the existence of a lifting of the M't/. space corresponding to a product X = TI {ai,b,} ieI and a product measure J.1= Q9 J.1i' with J.1i{a;}=J.1i{b,}=! for all iE/. ,eI Then, she reduced the general case to this one, via an isomorphism theorem concerning homogeneous measure algebras [87], [88]. A different and more direct proof of the existence of a lifting was subsequently given by the authors in [65]' A variant of this proof is presented in chapter 4.

Keywords

Finite Funktionaltransformation Lifting Morphism algebra mathematics product measure proof theorem

Authors and affiliations

  • A. Ionescu Tulcea
    • 1
  • C. Ionescu Tulcea
    • 1
  1. 1.Northwestern UniversityEvanstonUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-88507-5
  • Copyright Information Springer-Verlag Berlin Heidelberg 1969
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-88509-9
  • Online ISBN 978-3-642-88507-5
  • Series Print ISSN 0071-1136
  • Buy this book on publisher's site