R. A. Johnson showed that there is no translation-invariant Borel lifting for the measure algebra of 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. In this paper we study analogs of these results for category algebras (the Borel σ-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 α-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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm194-1-2,
author = {Maxim R. Burke},
title = {Invariant Borel liftings for category algebras of Baire groups},
journal = {Fundamenta Mathematicae},
volume = {193},
year = {2007},
pages = {15-44},
zbl = {1120.54025},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm194-1-2}
}
Maxim R. Burke. Invariant Borel liftings for category algebras of Baire groups. Fundamenta Mathematicae, Tome 193 (2007) pp. 15-44. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm194-1-2/