Before revision: We consider a generalization of persistence diagrams, namely Radon measures supported on the upper half plane for which we define natural extensions of Wasserstein and bottleneck distances between persistence diagrams . Such measures naturally appear in topological data analysis when considering continuous representations of persistence diagrams (e.g.  persistence surfaces)
After revision: We consider a generalization of persistence diagrams, namely Radon measures supported on the upper half plane . Such measures naturally appear in topological data analysis when considering continuous representations of persistence diagrams (e.g. persistence surfaces)