  We investigate the measures of dissipation and accretion related to the weak
solutions of the Camassa-Holm equation. Demonstrating certain properties of
nonunique characteristics, we prove a new representation formula for these
measures and conclude about their structural features, such us singularity with
respect to the Lebesgue measure. We apply these results to gain new insights
into the structure of weak solutions, proving in particular that measures of
accretion vanish for dissipative solutions of the Camassa-Holm equation.
