Measurable metrics, intrinsic metrics and Lipschitz functions

In the paper [29], N. Weaver introduced the notion of measurable metric and that of Lipschitz function with respect to such a metric. The study of these notions was pursued by the same author in subsequent papers ([30, 32, · · ·]) and in the book [31]. In our paper [16], we treated various important examples and, in particular, we studied the intrinsic measurable metric associated with a local Dirichlet form and notably the case of Wiener spaces. On the other hand, M. Hino and J.A. Ramirez ([15]) showed that the intrinsic measurable metric associated with a symmetric diffusion semigroup was strongly involved in the description of the Gaussian behavior in small time of such a semigroup. Moreover, we introduced in [17] the concentration function related to a measurable metric and studied the corresponding Gaussian concentration property. In this paper, we shall give a survey of this set of results.