  We show that the weak limit of the maximal measures for any degenerating
sequence of rational maps on the Riemann sphere must be a countable sum of
atoms. For a 1-parameter family f_t of rational maps, we refine this result by
showing that the measures of maximal entropy have a unique limit on the Riemann
sphere as the family degenerates. The family f_t may be viewed as a single
rational function on the Berkovich projective line over the completion of the
field of formal Puiseux series in t, and the limiting measure on the Riemann
sphere is the "residual measure" associated to the equilibrium measure on the
Berkovich line. For the proof, we introduce a new technique for quantizing
measures on the Berkovich projective line and demonstrate the uniqueness of
solutions to a quantized version of the pullback formula for the equilibrium
measure there.
