Published in Proceedings of the American Mathematical Society, Volume 123, Issue 3, March 1, 1995, pages 887-895.
This article was first published in Proceedings of the American Mathematical Society, published by the American Mathematical Society. Copyright © 1995 American Mathematical Society.
NOTE: At the time of publication, the author Theodore P. Hill was not yet affiliated with Cal Poly.
The definitive version is available at https://doi.org/10.1090/S0002-9939-1995-1233974-8.
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 σ-algebra on the positive reals, and results for invariant measures on the circle.