Invariant Borel liftings for category algebras of Baire groups
by
Maxim R. Burke
University of Prince Edward Island

R.A. Johnson showed that there is no translation-invariant Borel lifting for the measure algebra of the R/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. We study analogs of these results for category algebras (the Borel s-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 a-favorable, then the category algebra of G has no left invariant Borel lifting. 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 in [2].

[1] M.R. Burke, Invariant Borel liftings for category algebras of Baire groups, submitted.

[2] M.R. Burke, A.W. Miller, Models in which every nonmeager set is nonmeager in a nowhere dense Cantor set, Canad. J. Math. 57 (2005) 1139-1154.

Date received: February 28, 2006