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: Despite the obvious similarities between the metrics used in topological data analysis and those of optimal transport, an optimal-transport based formalism to study persistence diagrams and similar topological descriptors has yet to come. In this article, by considering the space of persistence diagrams as a measure space, and by observing that its metrics can be expressed as solutions of optimal partial transport problems, we introduce 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)