  We prove a Liouville property for uniformly almost localized (up to
translations) H 1-global solutions of the Camassa-Holm equation with a momentum
density that is a non negative finite measure. More precisely, we show that
such solution has to be a peakon. As a consequence, we prove that peakons are
asymptotically stable in the class of H 1-functions with a momentum density
that belongs to M + (R). Finally, we also get an asymptotic stability result
for train of peakons.
