Intrinsic metrics and Lipschitz functions

We study the notions of measurable metric and Lipschitz function which were introduced by N. Weaver ([12]), in the framework of Dirichlet spaces. To this respect, we bring some precisions and complements to [15], notably concerning links with the notion of intrinsic metric ([2]). In the particular case of an abstract Wiener space, we establish the relationship between these notions and that of H-metric ([5]) and μ-a.e. H-Lipschitz continuous function ([4]).