B. Kadane, Joseph et Jin, Jiashun, Uniform Distributions on the Integers: A connection to the Bernouilli Random Walk, Econometric reviews , 33(1-4), 2014, pp. 372-378
Associate to each subset of the integers its almost sure limiting relative frequency under the Bernouilli random walk, if it has one. The resulting probability space is purely finitely additive, and uniform in the sense of residue classes and shift-invariance. However, it is not uniform in the sense of limiting relative frequency.