(Rend. Acc. Naz. Sci. XL, Mem. Mat. stands for: Accademia Nazionale delle Scienze detta dei XL. Rendiconti. Serie V. Memorie di Matematica e Applicazioni. Parte I. Accad. Naz. Sci. XL, Rome. ISSN 0392-4106.)
In a 1995 paper by G. Jacopini, a sigma-algebra H of subsets of R is constructed, and a translation-invariant measure \nu extending the Lebesgue measure \lambda is defined on H, such that for each r in [0,1] there are E in H for which \nu(E\cap A) = r \lambda(A) for all Borel subsets A. Given these properties, it can be suggested, as an intuitive interpretation, that such sets as E are uniformly gray. The main purpose of this note is to discuss the extent to which such an intuitive characterization is reasonable for sets having the above properties. Also mentioned are some other, very similar results, which have been published elsewhere.