  Working within the framework of free actions of countable amenable groups on
compact metrizable spaces, we show that the small boundary property is
equivalent to a density version of almost finiteness, which we call almost
finiteness in measure, and that under this hypothesis the properties of almost
finiteness, comparison, and $m$-comparison for some nonnegative integer $m$ are
all equivalent. The proof combines an Ornstein-Weiss tiling argument with the
use of zero-dimensional extensions which are measure-isomorphic over singleton
fibres. These kinds of extensions are also employed to show that if every free
action of a given group on a zero-dimensional space is almost finite then so
are all free actions of the group on spaces with finite covering dimension.
Combined with recent results of Downarowicz-Zhang and
Conley-Jackson-Marks-Seward-Tucker-Drob on dynamical tilings and of
Castillejos-Evington-Tikuisis-White-Winter on the Toms-Winter conjecture, this
implies that crossed product C$^*$-algebras arising from free minimal actions
of groups with local subexponential growth on finite-dimensional spaces are
classifiable in the sense of Elliott's program. We show furthermore that, for
free actions of countably infinite amenable groups, the small boundary property
implies that the crossed product has uniform property $\Gamma$, which under
minimality confirms the Toms-Winter conjecture for such crossed products by the
aforementioned work of Castillejos-Evington-Tikuisis-White-Winter.
