  In this paper we define the magnitude of metric spaces using measures rather
than finite subsets as had been done previously and show that this agrees with
earlier work with Leinster in arXiv:0908.1582. An explicit formula for the
magnitude of an n-sphere with its intrinsic metric is given. For an arbitrary
homogeneous Riemannian manifold the leading terms of the asymptotic expansion
of the magnitude are calculated and expressed in terms of the volume and total
scalar curvature of the manifold. In the particular case of a homogeneous
surface the form of the asymptotics can be given exactly up to vanishing terms
and this involves just the area and Euler characteristic in the way conjectured
for subsets of Euclidean space in previous work.
