Haar null sets without $G_δ$ hulls

Let $G$ be an abelian Polish group, e.g. a separable Banach space. A subset $X \subset G$ is called Haar null (in the sense of Christensen) if there exists a Borel set $B \supset X$ and a Borel probability measure $μ$ on $G$ such that $μ(B+g)=0$ for every $g \in G$. 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 $G_δ$ Haar null set. We also show that $G_δ$ 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.

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