  We apply the Law of Total Probability to the construction of scale-invariant
probability distribution functions (pdfs), and require that probability
measures be dimensionless and unitless under a continuous change of scales. If
the scale-change distribution function is scale invariant then the constructed
distribution will also be scale invariant. Repeated application of this
construction on an arbitrary set of (normalizable) pdfs results again in
scale-invariant distributions. The invariant function of this procedure is
given uniquely by the reciprocal distribution, suggesting a kind of
universality. We separately demonstrate that the reciprocal distribution
results uniquely from requiring maximum entropy for size-class distributions
with uniform bin sizes.
