We apply the Law of Total Probability to the construction of scale-invariant pdf's, and require that probability measures be dimensionless and unitless under a continuous change of scales.

Iterating this procedure for an arbitrary set of normalized pdf's again produces scale-invariant distributions.

The invariant function of this iteration is given uniquely by the reciprocal distribution, suggesting a kind of universality.

Requiring maximum entropy for uniformly binned size-class distributions also leads uniquely to the reciprocal distribution.

We discuss some applications of the above to computation and to the evolution of genomes.