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Invariant Borel liftings for category algebras of Baire groups

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Burke M.

Fundamenta Mathematicae

2007

tom 194

nr 1

15-44

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.

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1 Fundamenta Mathematicae

1 Burke M.

1 2007

Wyniki wyszukiwania

Invariant Borel liftings for category algebras of Baire groups