I am reading N. H. Bingham's (2010) paper: ''Finite additivity vs. countable additivity''.

Page 8, he states:

"One area where the distinction between finite and countable additivity shows up most clearly is in the question of a uniform distribution over the integers. In the countably additive case, no such distribution can exist (the total mass would be infinity or zero depending on whether singletons had positive or zero measure). In the finitely additive case, such distributions do exist (all finite sets having zero measure)''.

I understand his point for the countable additive case, but do not understand why, in the finitely additive case, a uniform distribution over the integers exists. Could somebody explain? Thank you.

I am reading N. H. Bingham's (2010) paper: ''Finite additivity vs. countable additivity''.

Page 8, he states:

"One area where the distinction between finite and countable additivity shows up most clearly is in the question of a uniform distribution over the integers. In the countably additive case, no such distribution can exist (the total mass would be infinity or zero depending on whether singletons had positive or zero measure). In the finitely additive case, such distributions do exist (all finite sets having zero measure)''.

I understand his point for the countable additive case, but do not understand why, in the finitely additive case, a uniform distribution over the integers exists. Could somebody explain? Thank you.