  Motivated by giving a meaning to "The probability that a random integer has
initial digit d", we define a URI-set as a random set E of natural integers
such that each n>0 belongs to E with probability 1/n, independently of other
integers. This enables us to introduce two notions of densities on natural
numbers: The URI-density, obtained by averaging along the elements of E, and
the local URI-density, which we get by considering the k-th element of E and
letting k go to infinity. We prove that the elements of E satisfy Benford's
law, both in the sense of URI-density and in the sense of local URI-density.
Moreover, if b_1 and b_2 are two multiplicatively independent integers, then
the mantissae of a natural number in base b_1 and in base b_2 are independent.
Connections of URI-density and local URI-density with other well-known notions
of densities are established: Both are stronger than the natural density, and
URI-density is equivalent to log-density. We also give a stochastic
interpretation, in terms of URI-set, of the H_\infty-density.
