## Manifolds with Density

Spaces *M* with metrics and measures, so-called metric measure spaces or mm spaces (see e.g. Gromov [G]), include very general singular manifolds and weighted graphs. In the smooth case, *M* is a Riemannian manifold endowed with a smooth positive function or “density” *f*; the prescribed measure is just *f* times the Riemannian volume. In freshman calculus one studies surfaces and solids of revolution via their generating curves and regions in the halfplane {*x*>0} with density *f*(*x*) = 2π*x*. All quotient manifolds of Riemannian manifolds and homogeneous spaces *G/K* are Riemannian manifolds with density, and mm spaces were previously called spaces of homogeneous type (see [CW, pp. 587, 591]). Another example, long important to probabilists, is Euclidean space with Gaussian density.