  Let $\mathcal G_n$ denote the space of $n$-generated marked groups. We prove
that, for every $n\ge 2$, there exist $2^{\aleph_0}$ non-atomic,
$Out(F_n)$-invariant, mixing probability measures on $\mathcal G_n$. On the
other hand, there are non-empty closed subsets of $\mathcal G_n$ that admit no
$Out(F_n)$-invariant probability measure. Acylindrical hyperbolicity of the
group $Aut(F_n)$ plays a crucial role in the proof of both results. We also
discuss model theoretic implications of the existence of $Out(F_n)$-invariant,
ergodic probability measures on $\mathcal G_n$.
