Nonnegative measure-valued solutions of the continuity equation are uniquely
determined by their initial condition, if the characteristic ODE associated to
the velocity field has a unique solution. In this paper we give a partial
extension of this result to signed measure-valued solutions, under a
quantitative two-sided Osgood condition on the velocity field. Our results
extend those obtained for log-Lipschitz vector fields by Bahouri and Chemin
