*Accepted Paper*

**Inserted:** 22 sep 2020

**Last Updated:** 24 jul 2022

**Journal:** Probability Theory and Related Fields

**Year:** 2020

**Abstract:**

Inspired by the recent theory of Entropy-Transport problems and by the $\mathbf{D}$-distance of Sturm on \emph{normalised} metric measure spaces, we define a new class of complete and separable distances between metric measure spaces of possibly different total mass.

We provide several explicit examples of such distances, where a prominent role is played by a geodesic metric based on the Hellinger-Kantorovich distance. Moreover, we discuss some limiting cases of the theory, recovering the ``pure transport'' $\mathbf{D}$-distance and introducing a new class of ``pure entropic'' distances.

We also study in detail the topology induced by such Entropy-Transport metrics, showing some compactness and stability results for metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense.

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