Persistence of Rademacher-type and Sobolev-to-Lipschitz properties

We consider the Rademacher- and Sobolev-to-Lipschitz-type properties for arbitrary quasi-regular strongly local Dirichlet spaces. We discuss the persistence of these properties under localization, globalization, transfer to weighted spaces, tensorization, and direct integration. As byproducts, we obtain: necessary and sufficient conditions to identify a quasi-regular strongly local Dirichlet form on an extended metric topological σ-finite possibly non-Radon measure space with the Cheeger energy of the space; the tensorization of intrinsic distances; the tensorization of the Varadhan short-time asymptotics.