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

DOI:
https://doi.org/10.1090/S0002-9939-1993-1128726-1

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 translation-invariant lifting* *for* *is not universally measurable and does not have the Baire property*.

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DOI:
https://doi.org/10.1090/S0002-9939-1993-1128726-1

Article copyright:
© Copyright 1993
American Mathematical Society