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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 $ 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.


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DOI: https://doi.org/10.1090/S0002-9939-1993-1128726-1
Article copyright: © Copyright 1993 American Mathematical Society

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