An invariant mean on a discrete group 
is a positive linear functional 
on the vector
space of bounded functions 
with 
and
 |
for all 
and all such functions 
. This is the left-invariance convention.
Positivity means 
whenever 
,
and 1 denotes the constant function with value one (Juschenko
2015, pp. 28-31).
Positivity and normalization imply 
, where 
is the supremum norm,
so 
is bounded and has operator
norm one. It corresponds to a left-invariant finitely
additive probability measure on all subsets of 
by 
, where 
is the indicator function
of 
.
Conversely, such a finitely additive
probability measure defines 
first on simple functions
and then by extension in the supremum norm (Juschenko
2015, pp. 29-31).
A discrete group is amenable iff it admits an invariant mean. For a finite
group, an example is 
.
