  Using a generalized framework that consists of evolution of the solution to
the Camassa- Holm equation and its energy measure, we establish the
global-in-time orbital stability of peakons with respect to the perturbed
(energy) conservative solutions to the Camassa-Holm equation. Especially, we
extend the H1-stability result obtained by Constantin and Strauss (Comm. Pure
Appl. Math., 53(5), 603-610, 2000) globally-in-time, even after the perturbed
solutions experience wave breaking. In addition, our result also shows that the
singular part of the energy measure of the perturbed solutions will remain
stable for all times.
