Definition For Locally Compact Groups
Let be a locally compact Hausdorff group. Then it is well known that it possesses a unique, up-to-scale left- (or right-) rotation invariant ring (borel regular in the case of second countable) measure (left and right probability measure in the case of compact), the Haar measure. Consider the banach space of essentially-bounded measurable functions within this measure space (which is clearly independent of the scale of the Haar measure).
Definition 1. A linear functional is said to be a mean if has norm 1 and is non-negative (i.e. a.e. implies ).
Definition 2. A mean is said to be left-invariant (resp. right-invariant) if all with respect to the left (resp. right) shift action of (resp. ).
Definition 3. A locally compact hausdorff group is called amenable if it admits a left- (or right-)invariant mean.
Read more about this topic: Amenable Group
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