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Countable sections for locally compact group actions

  • Alexander S. Kechris (a1)
  • DOI:
  • Published online: 01 September 2008

It has been shown by J. Feldman, P. Hahn and C. C. Moore that every non-singular action of a second countable locally compact group has a countable (in fact so-called lacunary) complete measurable section. This is extended here to the purely Borel theoretic category, consisting of a Borel action of such a group on an analytic Borel space (without any measure). Characterizations of when an arbitrary Borel equivalence relation admits a countable complete Borel section are also established.

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[A]W. Ambrose . Representation of ergodic flows. Ann. of Math. 42 (1941), 723739.

[B]J. Burgess . A selection theorem for group actions. Pac. J. Math. 80 (1979), 333336.

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[FR]J. Feldman & A. Ramsay . Countable sections for free actions of groups. Adv. Math. 55 (1985), 224227.

[HKL]L. Harrington , A. S. Kechris & A. Louveau . A Glimm—Effros dichotomy for Borel equivalence relations. J. Amer. Math. Soc. 3 (1990), 903928.

[Ke1]A. S. Kechris . Measure and category in effective descriptive set theory. Ann. Math. Logic 5 (1973), 337384.

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[Var]V. S. Varadarajan . Groups of automorphisms of Borel spaces. Trans. Amer. Math. Soc. 109 (1963), 191220.

[W]V. M. Wagh . A descriptive version of Ambrose's representation theorem for flows. Proc. Ind. Acad. Sci. (Math. Sci.) 98 (1988), 101108.

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Ergodic Theory and Dynamical Systems
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