  We give a criterion for the weak convergence of unit Borel measures on the
N-dimensional Berkovich projective space over a complete non-archimedean field.
As an application, we give a sufficient condition for equidistribution in terms
of a strong Zariski-density property on the scheme-theoretic projective space
over the residue field. As a second application, in the case of residue
characteristic zero we give an ergodic-theoretic equidistribution result for
the powers of a point in the N-dimensional unit torus. This is a
non-archimedean analogue of a well-known complex equidistribution result of
Weyl, and its proof makes essential use of a theorem of Mordell-Lang type due
to Laurent.
