  Building on the first two authors' previous results, we prove a general
criterion for convergence of (possibly singular) Bergman measures towards
equilibrium measures on complex manifolds. The criterion may be formulated in
terms of growth properties of balls of holomorphic sections, or equivalently as
an asymptotic minimization of generalized Donaldson L-functionals. Our result
yields in particular the proof of a well-known conjecture in pluripotential
theory concerning the equidistribution of Fekete points, and it also gives the
convergence of Bergman measures towards equilibrium for Bernstein-Markov
measures. Applications to interpolation of holomorphic sections are also
discussed.
