The Kantorovich-Rubinstein duality theorem, proven in the case of a compact metric space in late 50s, stems from 18th century work of Monge on the transport of mass problem. The idea is as follows. Given probabilities \(P_ 1\) and \(P_ 2\) on a space S and a measurable cost function c(x,y) on \(S\times S\), one considers the so-called Wasserstein or Kantorovich-Rubinstein functional \[ {\hat \mu}_ c(P_ 1,P_ 2)=\inf \int_{S\times S}c(x,y)db(x,y), \] where the infimum is taken over all probabilities b on \(S\times S\) with marginals \(b_ 1=P_ 1\) and \(b_ 2=P_ 2\). When S is a finite set, then, by a linear programming result, \[ {\hat \mu}_ c(P_ 1,P_ 2)=\sup (\int f dP_ 1-\int g dP_ 2), \] where the supremum is taken over all functions on S satisfying a Lipschitz-type condition.
Initially, following some ideas of Dudley, the paper presents duality- type results in the case of separable metric spaces and cost functions that are not necessarily metrics. When S is the real line, the Wasserstein functional admits an elegant representation (Vallander’s formula) which is used later in a more general context related to work of Fortet and Mourier.
Further in the paper, rather general duality results are proved under fairly weak assumptions concerning the involved measures and a large class of cost functions c(x,y). The topology, compactness properties, and completeness, related to use of Wasserstein functional, are discussed in detail. The theory finds an application in the problem of approximation of a theoretical distribution by empirical distributions in terms of the functional \({\hat \mu}\).