Invariant Borel liftings for category algebras of Baire groups

Maxim R. Burke. "Invariant Borel liftings for category algebras of Baire groups." Fundamenta Mathematicae 194.1 (2007): 15-44. <http://eudml.org/doc/286157>.

@article{MaximR2007,
abstract = {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.},
author = {Maxim R. Burke},
journal = {Fundamenta Mathematicae},
keywords = {Baire space; lifting; Borel set; topological group},
language = {eng},
number = {1},
pages = {15-44},
title = {Invariant Borel liftings for category algebras of Baire groups},
url = {http://eudml.org/doc/286157},
volume = {194},
year = {2007},
}

TY - JOUR
AU - Maxim R. Burke
TI - Invariant Borel liftings for category algebras of Baire groups
JO - Fundamenta Mathematicae
PY - 2007
VL - 194
IS - 1
SP - 15
EP - 44
AB - 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.
LA - eng
KW - Baire space; lifting; Borel set; topological group
UR - http://eudml.org/doc/286157
ER -