Israel Journal of Mathematics

, Volume 51, Issue 4, pp 339–346 | Cite as

Measurable group actions are essentially Borel actions

  • Arlan Ramsay


If a locally compact groupG acts on a Lebesgue probability space (X, λ), it is natural to consider these conditions: (a) each group element preserves the class of λ, and (b) the action function is measurable. The latter is a weakening of the requirement that the action be Borel, providedX has a particular Borel structure as well as the σ-algebra of measurable sets. In this paper, we give an example showing that such an action need not be Borel relative to the given Borel structure, and prove that there is always a conull invariant subset and a new standard Borel structure on that subset for which the action is Borel. This is the meaning of the title.


Compact Group Lebesgue Space Borel Function Borel Space Measure Algebra 
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  1. 1.
    G. Birkhoff,Lattice Theory, 3rd ed., Vol. XXV, Amer. Math. Soc. Colloq. Publ., Providence, 1967.MATHGoogle Scholar
  2. 2.
    I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai,Ergodic Theory, Springer-Verlag, New York, 1982.MATHGoogle Scholar
  3. 3.
    G. W. Mackey,Induced representations of locally compact groups, I, Ann. Math.2 (1952), 101–139.MathSciNetCrossRefGoogle Scholar
  4. 4.
    G. W. Mackey,Borel structures in groups and their duals, Trans. Am. Math. Soc.85 (1957), 265–311.CrossRefMathSciNetGoogle Scholar
  5. 5.
    G. W. Mackey,Point realizations of transformation groups, Illinois J. Math.6 (1962), 327–335.MATHMathSciNetGoogle Scholar
  6. 6.
    J. von Neumann,Einige Sätze über messbare Abbildungen, Ann. Math.33 (1932), 574–586.CrossRefGoogle Scholar
  7. 7.
    G. K. Pedersen,C*-Algebras and their Automorphism Groups, London Math. Soc. Monographs No. 14 Academic Press, New York, 1979.Google Scholar
  8. 8.
    A. Ramsay,Virtual groups and group actions, Adv. Math.6 (1971), 253–322.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    A. Ramsay,Nontransitive quasiorbits in Mackey’s analysis of group extensions, Acta Math.137 (1976), 17–48.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    V. A. Rohlin,On the fundamental ideas of measure theory, Am. Math. Soc. Transl.10 (1962), 1–54.Google Scholar
  11. 11.
    P. Shields,The Theory of Bernoulli Shifts, University of Chicago Press, Chicago, 1973.MATHGoogle Scholar
  12. 12.
    B. Weiss,Measurable dynamics, Proceedings of the S. Kakutani Conference, to appear.Google Scholar

Copyright information

© The Weizmann Science Press of Israel 1985

Authors and Affiliations

  • Arlan Ramsay
    • 1
  1. 1.Department of MathematicsUniveristy of ColoradoBoulderUSA

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