Amenability for actions of étale groupoids on $C^{\ast }$-algebras and Fell bundles

We generalize Renault’s notion of measurewise amenability to actions of second countable, Hausdorff, étale groupoids on separable $C^{\ast }$-algebras and show that measurewise amenability characterizes nuclearity of the crossed product whenever the $C^{\ast }$-algebra acted on is nuclear. In the more general context of Fell bundles over second countable, Hausdorff, étale groupoids, we introduce a version of Exel’s approximation property. We prove that the approximation property implies nuclearity of the cross-sectional algebra whenever the unit bundle is nuclear. For Fell bundles associated to groupoid actions, we show that the approximation property implies measurewise amenability of the underlying action.