View Complete Reference

Bumby, R and Ellentuck, E (1969)

Finitely additive measures and the first digit problem

Fundamenta Mathematicae 65, pp. 33-42.

ISSN/ISBN: 0016-2736 DOI: Not available at this time.

Abstract: The first significant digit conjecture is stated as follows: The proportion of physical constants whose first significant digit lies between 1 and n, where 1 ≤ n ≤ 9 , is log10(n+1). In this connection the authors define various sets of finitely additive set functions defined on P(N), the power set of N, where N is the set of natural numbers, in order to find a ``reasonable'' class of measures for which the first significant digit conjecture for natural numbers would be probabilistically verified. ℳ is the set of non-atomic measures, i.e., those which satisfy the properties (i) µ(A ⋃ B)=µ(A)+µ(B) for A,B ⊂ N, A ⋂ B= ∅, (ii) µ(N)=1, (iii) µ({n})=0 for all n ∈ N. T consists of the translation invariant measures, which satisfy the additional property that µ(A)=µ(A+1) for µ ∈ T and for all A ⊂ N. If C is any class of measures and A ⊂ N, C(A) is defined to be the set { µ(A) | µ ∈ C}. The authors prove that if P is the set of natural numbers having first significant digit equal to 1, T(P) is the entire interval $[0,1]$, thereby showing that translation invariant measures are too general to settle the first significant digit problem. The authors then proceed to extend the measures contained in ℳ and T to the class S of sparse sets, which are the sets A of positive real numbers having the property that the cardinality of the set A ⋂ [n,n+1) is bounded for all n ∈ N. R, the class of scale invariant measures, is defined to consist of those µ ∈ T for which µ(A)= α µ( α A ) for every A ∈ S and every α >0. In other words, ''thinning'' the set A by multiplying each element in it by α has the ''reasonable'' effect of multiplying its measure by 1/α. A somewhat technical theorem is proved which immediately implies that if R is restricted to P(N) and if Pn is the set of natural numbers whose first significant digit lies between 1 and n, then for 1 ≤ n ≤ 9, S(Pn) is the singleton log10(n+1)} which verifies the conjecture for any µ ∈ R.

@article {MR0249386, AUTHOR = {Bumby, Richard and Ellentuck, Erik}, TITLE = {Finitely additive measures and the first digit problem}, JOURNAL = {Fund. Math.}, FJOURNAL = {Polska Akademia Nauk. Fundamenta Mathematicae}, VOLUME = {65}, YEAR = {1969}, PAGES = {33--42}, ISSN = {0016-2736}, MRCLASS = {10.50}, MRNUMBER = {0249386 (40 \#2631)}, MRREVIEWER = {M. C. Wunderlich}, URL = {}, }

Reference Type: Journal Article

Subject Area(s): Measure Theory