Haar null sets without G δ hulls

Abstract

Let G be an abelian Polish group, e.g., a separable Banach space. A subset X ⊂ G is called Haar null (in the sense of Christensen) if there exists a Borel set B ⊃ X and a Borel probability measure µ on G such that µ(B + g) = 0 for every g ∈ G. The term shy is also commonly used for Haar null, and co-Haar null sets are often called prevalent. Answering an old question of Mycielski we show that if G is not locally compact then there exists a Borel Haar null set that is not contained in any $${G_\delta }$$ Haar null set. We also show that $${G_\delta }$$ can be replaced by any other class of the Borel hierarchy, which implies that the additivity of the σ-ideal of Haar null sets is ω 1. The definition of a generalised Haar null set is obtained by replacing the Borelness of B in the above definition by universal measurability. We give an example of a generalised Haar null set that is not Haar null, more precisely, we construct a coanalytic generalised Haar null set without a Borel Haar null hull. This solves Problem GP from Fremlin’s problem list. Actually, all our results readily generalise to all Polish groups that admit a two-sided invariant metric.