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

Oliver Schirokauer and Joseph B. Kadane

Updated: February 2006


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.

Keywords:imit points, limiting relative frequency, non-conglomerability, probability charge, residue class, shift-invariance

Heidi Sestrich 2005-02-02
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