Haar null sets without Gδ hulls

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 Gδ Haar null set. We also show that Gδ 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 ω1. 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.