Liftings and the property of Baire in locally compact groups

Abstract

For each locally compact group G with Haar measure p , 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 \. If G is a-compact, H C G is a closed normal subgroup, and n: G —» G/H is the usual projection, then for each first category set A C G , there is a first category set E C G/H such that for each y e (G/H) E, Af)K~l(y) is a first category set relative to n~x(y). Theorem 2. // G is not discrete, then there is a Borel set E CG such that for any translation-invariant lifting p for (G, p), p(E) is not universally measurable and does not have the Baire property.