It's an obvious and well-known fact that there is no uniform probability measure on a set of natural numbers (i.e. the one that gives the same probability to each singleton).
On a recent probability seminar one professor mentioned that this nonexistance is one of the main objections to measure-theoretic formulation of probability and that even Kolmogorov admitted the problem in it.
What are possible solutions (i.e. alternative definitions of probability) to this problem?
I'm looking either for an informative answer or a good reference.



"Uniform probability" on a set of naturals
The correct notion here is **amenability**. This means that there is a positive linear form $\phi$ on the space of bounded functions that is invariant under translations:
$$\phi(\tau\_nf)=\phi(f),\qquad\forall f\in\ell^\infty(\mathbb Z),n\in\mathbb Z,$$
where $\tau\_nf(m):=f(m+n)$. As mentioned by Qiaochu, it is not countably additive, but only finitely additive (when applied to characteristic functions). It satisfies in addition $\phi({\bf 1})=1$. Of course, if $f$ has a finite support,then $\phi(f)=0$. Therefore, we must forget about "counting".


Not every group is amenable. Free groups with two or more generators are not.


**Edit**. As mentionned by Nate, one does not have an explicit form of $\phi$, because its existence follows from the axiom of choice. In particular, it is not unique.


