  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 $\mu$ on $G$ such
that $\mu(B+g)=0$ 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
$G_\delta$ 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
$\sigma$-ideal of Haar null sets is $\omega_1$.
  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.
