February, 2005

Tech Report

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)