Magnitude of metric measure spaces and integrals over geodesics

We propose a definition of magnitude for a length space with a Borel measure, which involves integrals over the set of geodesics. This quantity agrees with the magnitude of finite metric spaces, up to re-scaling the metric to ensure the convergence, when we use the counting measure on them. We also prove a version of the homogeneous magnitude theorem, by showing that the new definition agrees with the volume when we use the weight measure on a compact homogeneous Riemannian manifold. We compute various examples, which suggest that this quantity can capture information of non-uniqueness of geodesics, such as the injectivity radius, corresponding to the generating degrees of the magnitude homology.