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


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
Full Text: DOI


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