Liftings and the property of Baire in locally compact groups
Author:
Maxim R. Burke
Journal:
Proc. Amer. Math. Soc. 117 (1993), 10751082
MSC:
Primary 28A51; Secondary 28C10, 46G15, 54H05
MathSciNet review:
1128726
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Abstract: For each locally compact group with Haar measure , 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 is compact, is a closed normal subgroup, and is the usual projection, then for each first category set , there is a first category set such that for each is a first category set relative to . Theorem 2. If is not discrete, then there is a Borel set such that for any translationinvariant lifting for is not universally measurable and does not have the Baire property.
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 M. R. Burke and S. Shelah, Linear liftings for noncomplete probability spaces, Israel J. Math. (to appear). MR 1248919 (94m:03079)
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 P. R. Halmos, Measure theory, SpringerVerlag, New York, 1974. MR 0453532 (56:11794)
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 E. Hewitt and R. A. Ross, Abstract harmonic analysis. I, 2nd ed., SpringerVerlag, Berlin, 1979.
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 A. IonescuTulcea and C. IonescuTulcea, 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. 6397. MR 0212122 (35:2997)
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 K. Kuratowski, Topology, vol. I, Academic Press, New York, 1966. MR 0217751 (36:840)
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 J. Kupka and P. Prikry, Translationinvariant Borel liftings hardly ever exist, Indiana Univ. Math. J. 32 (1983), 717731. MR 711863 (85d:46061)
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 V. Losert, Some remarks on invariant liftings, Measure Theory Oberwolfach 1983 (D. Kölzow and D. MaharamStone, eds.), Lecture Notes in Math., vol. 1080, SpringerVerlag, Berlin, 1984, pp. 95110. MR 786689 (86g:28013)
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 J. C. Oxtoby, Measure and category, Academic Press, New York, 1971.
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 S. Shelah, Lifting problem of the measure algebra, Israel J. Math. 45 (1983), 9096. MR 710248 (85b:03092)
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 R. M. Solovay, A model of set theory in which every set of reals is Lebesgue measurable, Ann. of Math. (2)92 (1970), 156. MR 0265151 (42:64)
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 H. E. White, Topological spaces that are favorable for a player with perfect information, Proc. Amer. Math. Soc. 50 (1975), 477482. MR 0367941 (51:4183)
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DOI:
http://dx.doi.org/10.1090/S00029939199311287261
PII:
S 00029939(1993)11287261
Article copyright:
© Copyright 1993 American Mathematical Society
