(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.