Liftings and the property of Baire in locally compact groups

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.