Metric Measure Space

A metric measure space is a metric space equipped with a measure on its Borel sets. A probability metric measure space is a triple with . An equivalence of such spaces must preserve both distances and the measure, rather than just the metric. For a finite connected graph, graph distance and a probability measure on its vertices give an example. Random planar maps use the corner probability measure, which weights a vertex by its degree rather than assigning equal mass to every vertex. The Gromov-Hausdorff-Prokhorov distance compares compact probability metric measure spaces while retaining both structures.