The first digit problem and scale-invariance. (English) Zbl 0853.60002

Marcellini, Paolo (ed.) et al., Partial differential equations and applications. Collected papers in honor of Carlo Pucci on the occasion of his 70th birthday. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 177, 329-340 (1996).
The author’s purpose is to give a countably-additive measure-theoretic derivative of Benford’s law \[ \text{Prob[first significant digit(base }10)=k]= \log_{10} [(k+ 1)/ k], \quad k=1, 2, \dots, 9, \] [cf. F. Benford, Proc. Am. Philos. Soc. 78, 551-572 (1938; Zbl 0018.26502); R. A. Raimi, Am. Math. Monthly 83, 521-538 (1976; Zbl 0349.60014)] based on the classical scale-invariance hypothesis. By appropriate choice of measure space, the restriction to Borel subsets of \([1, 10)\), and use of arguments similar to the reviewer’s ones [Proc. Am. Math. Soc. 123, No. 3, 887-895 (1995; Zbl 0813.60002)], it is shown that scale-invariance of the probability measure implies absolute continuity, and Benford’s law then follows from the explicit expression of the distribution function and the resulting density.
For the entire collection see [Zbl 0834.00042].
Reviewer: T.P.Hill (Atlanta)


60A10 Probabilistic measure theory