  We show that every probability-measure-preserving action of a countable
amenable group G can be tiled, modulo a null set, using finitely many finite
subsets of G ("shapes") with prescribed approximate invariance so that the
collection of tiling centers for each shape is Borel. This is a dynamical
version of the Downarowicz--Huczek--Zhang tiling theorem for countable amenable
groups and strengthens the Ornstein--Weiss Rokhlin lemma. As an application we
prove that, for every countably infinite amenable group G, the crossed product
of a generic free minimal action of G on the Cantor set is Z-stable.
