  We define a metric in the space of positive finite positive measures that
extends the 2-Wasserstein metric, i.e. its restriction to the set of
probability measures is the 2-Wasserstein metric. We prove a dual and a dynamic
formulation and extend the gradient flow machinery of the Wasserstein space. In
addition, we relate the barycenter in this space to the barycenter in the
Wasserstein space of the normalized measures.
