We study a weak form of Gromov-Hausdorff convergence
of Riemannian manifolds, also known as
Benjamini-Schramm convergence. This concept
is also applicable to other areas and has widely
been studied in the context of graphs.
The main result is the continuity of characteristic
numbers normalized by the volume with respect
to the Benjamini-Schramm topology on the class
of Riemannian manifolds with a uniform lower
bound on injectivity radius and Ricci curvature.
An immediate consequence is a comparison theorem
that gives for any characteristic number a
linear bound in terms of the volume on the entire
class of manifolds mentioned. We give another
interpretation of the result showing that characteristic
numbers can be reconstructed with some
accuracy from local random information.
