We introduce the notion of an asymptotically invariant mean as a coarse averaging operator for a metric space and show that the existence of such an operator is equivalent to Yu’s property A. As an application we obtain a positive answer to Higson’s question concerning the existence of a cohomological characterisation of property A. Specifically we provide coarse analogues of group cohomology and bounded cohomology (controlled cohomology and asymptotically invariant cohomology, respectively) for a metric space X, and provide a cohomological characterisation of property A which generalises the results of Johnson and Ringrose describing amenability in terms of bounded cohomology. These results amplify Guentner’s observation that property A should be viewed as coarse amenability for a metric space. We further provide a generalisation of Guentner’s result that box spaces of a finitely generated group have property A if and only if the group is amenable. This is used to derive Nowak’s theorem that the union of finite cubes of all dimensions does not have property A
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