Uniform Distributions on the Natural Numbers

We compare the following three notions of uniformity for a finitely additive probability measure on the set of natural numbers: that it extend limiting relative frequency, that it be shift-invariant, and that it map every residue class mod m to 1/m. We find that these three types of uniformity can be naturally ordered. In particular, we prove that the set L of extensions of limiting relative frequency is a proper subset of the set S of shift-invariant measures and that S is a proper subset of the set R of measures which map residue classes uniformly. Moreover, we show that there are subsets G of N for which the range of possible values μ(G) for μ∈L is properly contained in the set of values obtained when μ ranges over S, and that there are subsets G which distinguish S and R analogously.