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We show that the weak limit of the maximal measures for any degenerating sequence of rational maps on the Riemann sphere 
${\hat{{\mathbb{C}}}} $ must be a countable sum of atoms. For a one-parameter family 
$f_t$ of rational maps, we refine this result by showing that the measures of maximal entropy have a unique limit on 
$\hat{{\mathbb{C}}}$ as the family degenerates. The family 
$f_t$ may be viewed as a single rational function on the Berkovich projective line 
$\mathbf{P}^1_{\mathbb{L}}$ over the completion of the field of formal Puiseux series in 
$t$ , and the limiting measure on 
$\hat{{\mathbb{C}}}$ is the ‘residual measure’ associated with the equilibrium measure on 
$\mathbf{P}^1_{\mathbb{L}}$ . 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 on 
$\mathbf{P}^1_{\mathbb{L}}$ .
