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Base-invariance implies Benford’s law. (English) Zbl 0813.60002
Summary: A derivation of Benford’s law or the first-digit phenomenon is given assuming only base-invariance of the underlying law. The only base- invariant distributions are shown to be convex combinations of two extremal probabilities, one corresponding to point mass and the other a log-Lebesgue measure. The main tools in the proof are identification of an appropriate mantissa $$\sigma$$-algebra on the positive reals, and results for invariant measures on the circle.

##### MSC:
 60A10 Probabilistic measure theory 28D05 Measure-preserving transformations
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