Finitely Additive Uniform Distributions on the Natural Numbers: Shift-Invariance

February, 2005

Tech Report

Author(s)

Oliver Schirokauer and Joseph B. Kadane

Abstract

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 $$\scriptstyle m$$ to $$\scriptstyle 1/m$$. We find that these three types of uniformity can be naturally ordered. In particular, we prove that the set $$\scriptstyle L$$ of extensions of limiting relative frequency is a proper subset of the set $$\scriptstyle S$$ of shift-invariant measures and that $$\scriptstyle S$$ is a proper subset of the set $$\scriptstyle R$$ of measures which map residue classes uniformly. Moreover, we show that there are subsets $$\scriptstyle G$$ of $$\scriptstyle\enn$$ for which the range of possible values $$\scriptstyle\mu(G)$$ for $$\scriptstyle \mu\in L$$ is properly contained in the set of values obtained when $$\scriptstyle \mu$$ ranges over $$\scriptstyle S$$, and that there are subsets $$\scriptstyle G$$ which distinguish $$\scriptstyle S$$ and $$\scriptstyle R$$ analogously.

(Revised 02/06)