Invariant Borel liftings for category algebras of Baire groups
Tom 194 / 2007
Streszczenie
R. A. Johnson showed that there is no translation-invariant Borel lifting for the measure algebra of $\mathbb R/\mathbb Z$ equipped with Haar measure, a result which was generalized by M. Talagrand to non-discrete locally compact abelian groups and by J. Kupka and K. Prikry to arbitrary non-discrete locally compact groups. In this paper we study analogs of these results for category algebras (the Borel $\sigma$-algebra modulo the ideal of first category sets) of topological groups. Our main results are for the class of non-discrete separable metric groups. We show that if $G$ in this class is weakly $\alpha$-favorable, then the category algebra of $G$ has no left-invariant Borel lifting. (This particular result does not require separability and implies a corresponding result for locally compact groups which are not necessarily metric.) Under the Continuum Hypothesis, many groups in the class have a dense Baire subgroup which has a left-invariant Borel lifting. On the other hand, there is a model in which the category algebra of a Baire group in the class never has a left-invariant Borel lifting. The model is a variation on one constructed by A. W. Miller and the author where every second category set of reals has a relatively second category intersection with a nowhere dense perfect set.
