Mesures invariantes par translation, classes de Dynkin first-digit problem. (Translation invariant measures; first-digit problem Dynkin classes) (Q581580)

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. \textit{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 \textit{L. E. Dubins} and \textit{D. Margolies} [Naturally integrable functions, Univ. of California, Berkeley, per. bibl.].