Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Liftings and the property of Baire in locally compact groups

Author: Maxim R. Burke
Journal: Proc. Amer. Math. Soc. 117 (1993), 1075-1082
MSC: Primary 28A51; Secondary 28C10, 46G15, 54H05
MathSciNet review: 1128726
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For each locally compact group $ G$ with Haar measure $ \mu $, we obtain the following results. The first is a version for group quotients of a classical result of Kuratowski and Ulam on first category subsets of the plane. The second is a strengthening of a theorem of Kupka and Prikry; we obtain it by a much simpler technique, building on work of Talagrand and Losert.

Theorem 1. If $ G$ is $ \sigma $-compact, $ H \subseteq G$ is a closed normal subgroup, and $ \pi :G \to G/H$ is the usual projection, then for each first category set $ A \subseteq G$, there is a first category set $ E \subseteq G/H$ such that for each $ y \in (G/H) - E,\;A \cap {\pi ^{ - 1}}(y)$ is a first category set relative to $ {\pi ^{ - 1}}(y)$.

Theorem 2. If $ G$ is not discrete, then there is a Borel set $ E \subseteq G$ such that for any translation-invariant lifting $ \rho $ for $ (G,\mu ),\;\rho (E)$ is not universally measurable and does not have the Baire property.

References [Enhancements On Off] (What's this?)

  • [BJ] M. R. Burke and J. Just, Liftings for Haar measure on $ {\{ 0,1\} ^\kappa }$, Israel J. Math. 73 (1991), 33-44. MR 1119925 (92h:03075)
  • [BS] M. R. Burke and S. Shelah, Linear liftings for non-complete probability spaces, Israel J. Math. (to appear). MR 1248919 (94m:03079)
  • [C] J. P. R. Christensen, Topology and Borel structure, North-Holland, Amsterdam, 1974. MR 0348724 (50:1221)
  • [H] P. R. Halmos, Measure theory, Springer-Verlag, New York, 1974. MR 0453532 (56:11794)
  • [HR] E. Hewitt and R. A. Ross, Abstract harmonic analysis. I, 2nd ed., Springer-Verlag, Berlin, 1979.
  • [I1] A. Ionescu-Tulcea and C. Ionescu-Tulcea, On the existence of a lifting commuting with the left translations of an arbitrary locally compact group, Proc. Fifth Berkeley Sympos. Math. Statist. and Probab. (Berkeley Calif. 1965/66), vol. II; Contributions to Probab. Theory, Part I, Univ. of Calif. Press, Berkeley, CA, 1967, pp. 63-97. MR 0212122 (35:2997)
  • [12] -, Topics in the theory of liftings, Springer-Verlag, New York, 1969.
  • [J] R. A. Johnson, Strong liftings which are not Borel liftings, Proc. Amer. Math. Soc. 80 (1980), 234-236. MR 577750 (81m:46061)
  • [Ju] W. Just, A modification of Shelah's oracle-cc with applications, Trans. Amer. Math. Soc. 329 (1992), 325-356. MR 1022167 (92j:03047)
  • [K] K. Kuratowski, Topology, vol. I, Academic Press, New York, 1966. MR 0217751 (36:840)
  • [KP] J. Kupka and P. Prikry, Translation-invariant Borel liftings hardly ever exist, Indiana Univ. Math. J. 32 (1983), 717-731. MR 711863 (85d:46061)
  • [L] V. Losert, Some remarks on invariant liftings, Measure Theory Oberwolfach 1983 (D. Kölzow and D. Maharam-Stone, eds.), Lecture Notes in Math., vol. 1080, Springer-Verlag, Berlin, 1984, pp. 95-110. MR 786689 (86g:28013)
  • [M] D. Maharam, On a theorem of von Neumann, Proc. Amer. Math. Soc. 9 (1958), 987-994. MR 0105479 (21:4220)
  • [O] J. C. Oxtoby, Measure and category, Academic Press, New York, 1971.
  • [S] S. Shelah, Lifting problem of the measure algebra, Israel J. Math. 45 (1983), 90-96. MR 710248 (85b:03092)
  • [So] R. M. Solovay, A model of set theory in which every set of reals is Lebesgue measurable, Ann. of Math. (2)92 (1970), 1-56. MR 0265151 (42:64)
  • [T] M. Talagrand, La pathologie des relèvements invariants, Proc. Amer. Math. Soc. 84 (1982), 379-382. MR 640236 (83a:46051)
  • [W] H. E. White, Topological spaces that are $ \alpha $-favorable for a player with perfect information, Proc. Amer. Math. Soc. 50 (1975), 477-482. MR 0367941 (51:4183)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 28A51, 28C10, 46G15, 54H05

Retrieve articles in all journals with MSC: 28A51, 28C10, 46G15, 54H05

Additional Information

Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society