A magnitude for metric measure spaces is defined using geodesic integrals; it recovers finite-space magnitude (rescaled) and manifold volume in special cases, and appears sensitive to geodesic non-uniqueness.
Dedicata168 (2014), 291–310
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Magnitude of metric measure spaces and integrals over geodesics
A magnitude for metric measure spaces is defined using geodesic integrals; it recovers finite-space magnitude (rescaled) and manifold volume in special cases, and appears sensitive to geodesic non-uniqueness.