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Measurable group actions are essentially Borel actions

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Abstract

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.

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References

  1. G. Birkhoff,Lattice Theory, 3rd ed., Vol. XXV, Amer. Math. Soc. Colloq. Publ., Providence, 1967.

    MATH  Google Scholar 

  2. I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai,Ergodic Theory, Springer-Verlag, New York, 1982.

    MATH  Google Scholar 

  3. G. W. Mackey,Induced representations of locally compact groups, I, Ann. Math.2 (1952), 101–139.

    Article  MathSciNet  Google Scholar 

  4. G. W. Mackey,Borel structures in groups and their duals, Trans. Am. Math. Soc.85 (1957), 265–311.

    Article  MathSciNet  Google Scholar 

  5. G. W. Mackey,Point realizations of transformation groups, Illinois J. Math.6 (1962), 327–335.

    MATH  MathSciNet  Google Scholar 

  6. J. von Neumann,Einige Sätze über messbare Abbildungen, Ann. Math.33 (1932), 574–586.

    Article  Google Scholar 

  7. G. K. Pedersen,C*-Algebras and their Automorphism Groups, London Math. Soc. Monographs No. 14 Academic Press, New York, 1979.

    Google Scholar 

  8. A. Ramsay,Virtual groups and group actions, Adv. Math.6 (1971), 253–322.

    Article  MATH  MathSciNet  Google Scholar 

  9. A. Ramsay,Nontransitive quasiorbits in Mackey’s analysis of group extensions, Acta Math.137 (1976), 17–48.

    Article  MATH  MathSciNet  Google Scholar 

  10. V. A. Rohlin,On the fundamental ideas of measure theory, Am. Math. Soc. Transl.10 (1962), 1–54.

    Google Scholar 

  11. P. Shields,The Theory of Bernoulli Shifts, University of Chicago Press, Chicago, 1973.

    MATH  Google Scholar 

  12. B. Weiss,Measurable dynamics, Proceedings of the S. Kakutani Conference, to appear.

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Authors and Affiliations

  1. Department of Mathematics, Univeristy of Colorado, 80309, Boulder, CO, USA

    Arlan Ramsay

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  1. Arlan Ramsay
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Supported by a University of Colorado Faculty Fellowship.

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Ramsay, A. Measurable group actions are essentially Borel actions. Israel J. Math. 51, 339–346 (1985). https://doi.org/10.1007/BF02764724

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  • DOI: https://doi.org/10.1007/BF02764724

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