Distances on probability measures and random variables

In this paper we lift fundamental topological structures on probability measures and random variables, in particular the weak topology, convergence in law and finite-dimensional convergence to an isometric level. This allows for an isometric quantitative study of important concepts such as relative compactness, tightness, stochastic equicontinuity, Prohorov's theorem and σ-smoothness. In doing so we obtain numerical results which allow for the development of an intrinsic approximation theory and from which moreover all classical topological results follow as easy corollaries.