  We prove that a Ricci curvature based method of triangulation of compact
Riemannian manifolds, due to Grove and Petersen, extends to the context of
weighted Riemannian manifolds and more general metric measure spaces. In both
cases the role of the lower bound on Ricci curvature is replaced by the
curvature-dimension condition ${\rm CD}(K,N)$. We show also that for weighted
Riemannian manifolds the triangulation can be improved to become a thick one
and that, in consequence, such manifolds admit weight-sensitive
quasimeromorphic mappings. An application of this last result to information
manifolds is considered.
  Further more, we extend to weak ${\rm CD}(K,N)$ spaces the results of Kanai
regarding the discretization of manifolds, and show that the volume growth of
such a space is the same as that of any of its discretizations.
