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 log_{10}(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 P_{n} is the set of natural numbers whose first significant digit lies between 1 and n, then for 1 ≤ n ≤ 9, S(P_{n}) is the singleton log_{10}(n+1)} which verifies the conjecture for any µ ∈ R.
Bibtex:
@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 = {https://www.impan.pl/en/publishing-house/journals-and-series/fundamenta-mathematicae/all/65/1/97004/finitely-additive-measures-and-the-first-digit-problem},
}
Reference Type: Journal Article
Subject Area(s): Measure Theory