Invariant Borel liftings for category algebras of Baire groups

Maxim R. Burke¹

¹ Department of Mathematics and Statistics University of Prince Edward Island Charlottetown, PE, Canada C1A 4P3

Fundamenta Mathematicae, Tome 194 (2007) no. 1, pp. 15-44

Résumé

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.

Détail

DOI : 10.4064/fm194-1-2

Keywords: nbsp johnson showed there translation invariant borel lifting measure algebra mathbb mathbb equipped haar measure result which generalized nbsp talagrand non discrete locally compact abelian groups nbsp kupka nbsp prikry arbitrary non discrete locally compact groups paper study analogs these results category algebras borel sigma algebra modulo ideal first category sets topological groups main results class non discrete separable metric groups class weakly alpha favorable category algebra has left invariant borel lifting particular result does require separability implies corresponding result locally compact groups which necessarily metric under continuum hypothesis many groups class have dense baire subgroup which has left invariant borel lifting other there model which category algebra baire group class never has left invariant borel lifting model variation constructed nbsp miller author where every second category set reals has relatively second category intersection nowhere dense perfect set