Pyramidal Compactification of Asymmetric Metric Measure Spaces via Adjoint Transport

Abstract

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.

Disclosure

“34 SHIGEAKI YOKOTA Acknowledgments The author would like to thank Professor Takashi Shioya for many helpful sug- gestions and guidance. The author used Claude and GPT-5.6-series Codex models as AI-assisted tools in preparing this manuscript. The author reviewed and revised the mathematical content and takes full responsibility for the final manuscript. R”

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2026-08-23T19:19:03.817253Z

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