The notion of topological amenability of a locally compact groupoid G endowed with a Haar system was first introduced by the author as a convenient sufficient condition for measurewise amenability.
The notion of Borel amenability for a groupoid, used by the author, is that of the 1-amenability in the case of countable Borel equivalence relations. The main result is: “A \(\sigma\)-compact locally compact groupoid possessing a continuous Haar system is topologically amenable if and only if it is Borel amenable.”
Examples and applications are given.