On a characterization of $N$-$\mathcal F$-amenability

We fix some issues in the proof of the characterization theorem of N -Famenability due to Guentner, Willett and Yu.1. Introduction.The notion of N -F-amenability was introduced by Bartels, Lck and Reich [7, Theorem 1.2], [6, Assumption 1.4] to prove the Farrell-Jones conjecture for hyperbolic groups (see [3], [10] for the Farrell-Jones conjecture and its applications).The terminology "N -F-amenable" is due to [2, Definition 0.1].In [9, Definition 4.6], being N -F-amenable is said to be N -BLR for F.Let X be an action of a discrete group on a compact Hausdorff space X by homeomorphisms, F a non-empty collection of subgroups of which is closed under conjugations and taking subgroups, and N a nonnegative integer.N -F-amenability is defined as follows (see Section 2 for undefined notations and terminology):Definition 1.1.The action X is said to be N -F-amenable (or N -BLR for F) if for every finite subset E of there exists a -equivariant open cover U of X satisfying the following properties:The smallest integer N such that X is N -F-amenable is also called the equivariant asymptotic dimension of X with respect to F [13, 2020