  In this paper, we address the problem of constructing a uniform probability
measure on $\mathbb{N}$. Of course, this is not possible within the bounds of
the Kolmogorov axioms and we have to violate at least one axiom. We define a
probability measure as a finitely additive measure assigning probability $1$ to
the whole space, on a domain which is closed under complements and finite
disjoint unions. We introduce and motivate a notion of uniformity which we call
weak thinnability, which is strictly stronger than extension of natural
density. We construct a weakly thinnable probability measure and we show that
on its domain, which contains sets without natural density, probability is
uniquely determined by weak thinnability. In this sense, we can assign uniform
probabilities in a canonical way. We generalize this result to uniform
probability measures on other metric spaces, including $\mathbb{R}^n$.
