  In this article, we generalize the Wasserstein distance to measures with
different masses. We study the properties of such distance. In particular, we
show that it metrizes weak convergence for tight sequences.
  We use this generalized Wasserstein distance to study a transport equation
with source, in which both the vector field and the source depend on the
measure itself. We prove existence and uniqueness of the solution to the Cauchy
problem when the vector field and the source are Lipschitzian with respect to
the generalized Wasserstein distance.
