Pyramidal Compactification of Asymmetric Metric Measure Spaces via Adjoint Transport

A quasi-metric measure space (qm-space) is a set with a directed distance whose symmetrization is a complete separable metric, together with a Borel probability measure of full support. Motivated by the problem of determining the pyramid limits of beta measures on forward Funk balls, we construct a compact metric space of pyramids of qm-spaces. The associated-pyramid map from the concentration-distance space of qm-spaces into this compact space is a $1$-Lipschitz topological embedding with dense image. As a secondary result, we prove that the box-distance space of qm-spaces is complete and separable.