  In this paper we study stable finiteness of ample groupoid algebras with
applications to inverse semigroup algebras and Leavitt path algebras,
recovering old results and proving some new ones. In addition, we develop a
theory of (faithful) traces on ample groupoid algebras, mimicking the
$C^\ast$-algebra theory but taking advantage of the fact that our functions are
simple and so do not have integrability issues, even in the non-Hausdorff
setting. The theory of traces is closely connected with the theory of invariant
means on Boolean inverse semigroups. We include an appendix on stable
finiteness of more general semigroup algebras, improving on an earlier result
of Munn, which is independent of the rest of the paper.
