Liftings and the Property of Baire in Locally Compact Groups

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.