An amenable group, in the discrete setting, is a group 
whose bounded
functions 
admit a positive normalized left-invariant mean.
This is a linear functional 
on those functions with 
, 
whenever 
, and 
for every 
.
Equivalently, an amenable discrete group admits a left-invariant finitely additive probability measure on all its subsets with total mass one. Finite groups have such a measure by normalized counting.
For a countably infinite set underlying 
, no left-invariant probability
measure can exist on all subsets. Invariance would
give every singleton set the same measure.
That value must be zero by finite additivity,
but countable additivity would then give
total measure
zero. This explains why finite additivity is
used in the characterization above (Juschenko 2015, pp. 28-31).
Factors.