## Mesures invariantes par translation, classes de Dynkin first-digit problem. (Translation invariant measures; first-digit problem Dynkin classes).(French)Zbl 0627.28010

In the first 4 sections of this long and interesting paper the authors study asymptotic and analytic densities of some subsets of $${\mathbb{N}}^*$$ by means of the notion of finitely additive measures on algebras. They also give a solution to the first-digit problem [cf., e.g. B. J. Flehinger, Am. Math. Mon. 73, 1056-1061 (1966; Zbl 0147.175)]. In Sections 6 and 7, they present a general theory of $$\alpha$$-classes (algebras of subsets of $${\mathbb{N}}^*$$ which are stable under translation and “homothety”) and measures over these $$\alpha$$-classes. In the last two sections, a link is established between the notion of natural integrability and the notion of a net of probability measures over $${\mathcal P}({\mathbb{N}}^*)$$, which are $$\sigma$$-additive and asymptotically invariant under translation. Finally, a generalization is obtained of the criterion of natural integrability of L. E. Dubins and D. Margolies [Naturally integrable functions, Univ. of California, Berkeley, per. bibl.].
Reviewer: A.N.Philippon

### MSC:

 28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures 60A10 Probabilistic measure theory 11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. 11B83 Special sequences and polynomials

Zbl 0147.175
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### References:

 [1] Ayoub, R, An introduction to analytic theory of numbers, (1963), Amer. Math. Soc Providence, R. I · Zbl 0128.04303 [2] Bauer, H, Wahrscheinlichkeitsrechnung und grundzüge der masstheorie, (1964), Sammlung Göschen Berlin · Zbl 0211.20102 [3] {\scL. E. Dubins and D. Margolies}, “Naturally Integrable Functions” Univ. of California, Berkeley. · Zbl 0461.28008 [4] Diaconis, P, Weak and strong averages in probability and the theory of numbers, () [5] Diaconis, P; Stein, C, Some Tauberian theorems related to coin tossing, Ann. probab., 6, 3, 483-490, (1978) · Zbl 0396.10044 [6] Dynkin, E.B, Die grundlagen der theorie der markoffschen prozesse, (1961), Berlin/Göttingen/Heidelberg · Zbl 0091.13604 [7] Flehinger, B.J, On the probability that a random integer has initial digit A, Amer. math. monthly, 73, 1056-1061, (1966) · Zbl 0147.17502 [8] Greenleaf, F.P, Invariant measures on topological groups, (1969), Van Nostrand-Reinhold Princeton, N. J · Zbl 0174.19001 [9] Hardy, G.H; Wright, E.M, An introduction to the theory of numbers, (1968), Oxford Univ. Press, (Clarendon) London/New York · Zbl 0020.29201 [10] Kubilius, J, Probabilistic methods in the theory of numbers, (1964), Amer. Math. Soc Providence, R. I · Zbl 0133.30203 [11] Nanopoulos, Ph, Loi de Dirichlet sur $$N$$∗ et pseudo probabilité, C. R. acad. sci., 280, 1543-1546, (1975) · Zbl 0309.60007 [12] Nanopoulos, Ph, Sur LES distributions limites de fonctions arithmétiques additives, () [13] Nymann, J.E, Amer. math. monthly, 79, 63, (1972) [14] Raimi, R.A, The first digit problem, Amer. math. monthly, 83, 521-538, (1976) · Zbl 0349.60014 [15] Scozzafava, R, Un esempio concreto di probabilità non σ-additiva: la distribuzione Della prima cifra significativa dei dati statistici, Boll. un. mat. ital., 18, 5, 403-410, (1981) · Zbl 0467.62008
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