Various notions of amenability for not necessarily locally compact groupoids

We start with a groupoid G endowed with a family W of subsets mimicking the properties of a neighborhood basis of the unit space (of a topological groupoid with paracompact unit space). Using the family W we endow each G-space with a uniform structure. The uniformities of the G-spaces allow us to define various notions of amenability for the G-equivariant maps. As in [C. Anantharaman-Delaroche and J. Renault, Amenable Groupoids. Monographie de L'Enseignement Mathematique No 36, Geneve, 2000], the amenability of the groupoid G is defined as the amenability of its range map. If the groupoid G is a group, all notions of amenability that we introduce coincide with the classical notion of amenability for topological (not necessarily locally-compact) groups.