**Oliver Schirokauer and Joseph B. Kadane**

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

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

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