A norm for signed measures, with application to non local transport equation with source term

We provide a norm on the space of signed Radon measures with finite mass, based on the generalized Wasserstein distance for measures of different mass. We explain precisely what is a converging sequence for this norm, in particular, it may be not uniformly bounded and not uniformly tight. This norm enables us to obtain existence and uniqueness of non local and non linear transport equations with source term when initial condition is a signed measure, as the unique limit of a convergent scheme.