A statistical test of isomorphism between metric-measure spaces using the distance-to-a-measure signature

Revision of a fuzzy distance measure

A note on the Gromov - Hausdorff - Prokhorov distance between ( locally ) compact metric measure spaces ∗

We present an extension of the Gromov-Hausdorff metric on the set of compact metric spaces: the Gromov-Hausdorff-Prokhorov metric on the set of compact metric spaces endowed with a finite measure. We then extend it to the non-compact case by describing a metric on the set of rooted complete locally compact length spaces endowed with a boundedly finite measure. We prove that this space with the ...

Marked Metric Measure Spaces

A marked metric measure space (mmm-space) is a triple (X , r,μ), where (X , r) is a complete and separable metric space and μ is a probability measure on X × I for some Polish space I of possible marks. We study the space of all (equivalence classes of) marked metric measure spaces for some fixed I . It arises as a state space in the construction of Markov processes which take values in random ...

Some properties of a new fuzzy distance measure

Quasi-metric Spaces with Measure

The phenomenon of concentration of measure on high dimensional structures is usually stated in terms of a metric space with a Borel measure, also called an mm-space. We extend some of the mm-space concepts to the setting of a quasi-metric space with probability measure (pq-space). Our motivation comes from biological sequence comparison: we show that many common similarity measures on biologica...