Before revision:  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.
After revision:  In this article, by considering the space of persistence diagrams as a space of discrete measures , 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.