  We propose a conjectural framework for computing Gorenstein measures and
stringy Hodge numbers in terms of motivic integration over arcs of smooth Artin
stacks, and we verify this framework in the case of fantastacks, which are
certain toric Artin stacks that provide (non-separated) resolutions of
singularities for toric varieties. Specifically, let $\mathcal{X}$ be a smooth
Artin stack admitting a good moduli space $\pi: \mathcal{X} \to X$, and assume
that $X$ is a variety with log-terminal singularities, $\pi$ induces an
isomorphism over a nonempty open subset of $X$, and the exceptional locus of
$\pi$ has codimension at least 2. We conjecture a formula for the motivic
measure for $\mathcal{X}$ in terms of the Gorenstein measure for $X$ and a
function measuring the degree to which $\pi$ is non-separated. We also
conjecture that if the stabilizers of $\mathcal{X}$ are special groups in the
sense of Serre, then almost all arcs of $X$ lift to arcs of $\mathcal{X}$, and
we explain how in this case, our conjectures imply a formula for the stringy
Hodge numbers of $X$ in terms of a certain motivic integral over the arcs of
$\mathcal{X}$. We prove these conjectures in the case where $\mathcal{X}$ is a
fantastack.
