Hostname: page-component-f7d5f74f5-z2nk8 Total loading time: 0 Render date: 2023-10-04T22:55:41.066Z Has data issue: false Feature Flags: { "corePageComponentGetUserInfoFromSharedSession": true, "coreDisableEcommerce": false, "coreDisableSocialShare": false, "coreDisableEcommerceForArticlePurchase": false, "coreDisableEcommerceForBookPurchase": false, "coreDisableEcommerceForElementPurchase": false, "coreUseNewShare": true, "useRatesEcommerce": true } hasContentIssue false

Countable sections for locally compact group actions

Published online by Cambridge University Press:  19 September 2008

Alexander S. Kechris
Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA


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.

Research Article
Copyright © Cambridge University Press 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)



[A]Ambrose, W.. Representation of ergodic flows. Ann. of Math. 42 (1941), 723739.CrossRefGoogle Scholar
[B]Burgess, J.. A selection theorem for group actions. Pac. J. Math. 80 (1979), 333336.CrossRefGoogle Scholar
[C]Christensen, J. P. R.. Topology and Borel Structure. (North-Holland, Amsterdam, 1974).Google Scholar
[FHM]Feldman, J., Hahn, P. & Moore, C. C.. Orbit structure and countable sections for actions of continuous groups. Adv. Math. 26 (1979), 186230.Google Scholar
[FM]Feldman, J. & Moore, C. C.. Ergodic equivalence relations, cohomology and von Neumann algebras, I. Trans. Amer. Math. Soc. 234 (1977), 289324.CrossRefGoogle Scholar
[FR]Feldman, J. & Ramsay, A.. Countable sections for free actions of groups. Adv. Math. 55 (1985), 224227.CrossRefGoogle Scholar
[F]Forrest, P. H.. Virtual subgroups of ℝ″ and ℤ″. Adv. Math. 3 (1974), 187207.Google Scholar
[HKL]Harrington, L., Kechris, A. S. & Louveau, A.. A Glimm—Effros dichotomy for Borel equivalence relations. J. Amer. Math. Soc. 3 (1990), 903928.CrossRefGoogle Scholar
[Ke1]Kechris, A. S.. Measure and category in effective descriptive set theory. Ann. Math. Logic 5 (1973), 337384.CrossRefGoogle Scholar
[Ke2]Kechris, A. S.. The structure of Borel equivalence relations in Polish spaces. Set Theory and the Continuum. Judah, H., Just, W. and Woodin, W. H., eds, MSRI Publications, Springer-Verlag, to appear.Google Scholar
[Ku]Kuratowski, K.. Topology. Vol. I (Academic Press: New York, 1966).Google Scholar
[Ma]Mackey, G. W.. Borel structures in groups and their duals. Trans. Amer. Math. Soc. 85 (1957), 134165.CrossRefGoogle Scholar
[Mi]Miller, D.. On the measurability of orbits in Borel actions. Proc. Amer. Math. Soc. 63 (1977), 165170.CrossRefGoogle Scholar
[MZ]Montgomery, D. & Zippin, L.. Topological Transformation groups (Interscience: New York, 1955).Google Scholar
[Mo]Moschovakis, Y. N.. Descriptive Set Theory. (North-Holland: Amsterdam, 1980).Google Scholar
[R1]Ramsay, A.. Topologies on measured groupoids. J. Fund. Anal. 47 (1982), 314343.CrossRefGoogle Scholar
[R2]Ramsay, A.. Local product structure for group actions. Ergod. Th. & Dynam. Sys. 11 (1991), 209217.CrossRefGoogle Scholar
[Var]Varadarajan, V. S.. Groups of automorphisms of Borel spaces. Trans. Amer. Math. Soc. 109 (1963), 191220.CrossRefGoogle Scholar
[Vau]Vaught, R. L.. Invariant sets in topology and logic. Fund. Math. 82 (1974), 269283.CrossRefGoogle Scholar
[W]Wagh, V. M.. A descriptive version of Ambrose's representation theorem for flows. Proc. Ind. Acad. Sci. (Math. Sci.) 98 (1988), 101108.CrossRefGoogle Scholar