1.1 Mass Transportation Problems in Probability Theory This chapter provides a basic introduction to mass transportation pr- lems (MTPs). We introduce some of the methods used in studying MTPs: dualrepresentations,explicitsolutions,topologicalproperties.Weshallalso discuss some applications of MTPs. The following measure-theoretic problems are well-known continuous casesofMKPs(see,forexample,Dudley(1976),LevinandMilyutin(1979), R¨ uschendorf (1979, 1981), Kemperman (1983), Kellerer (1984), Rachev (1984b, 1991c) and the references therein). TheMonge–Kantorovichmasstransportationproblem(MKP):Given?xed probability measuresP andP on a separable metric spaceS and a mea- 1 2 surable cost functionc on the Cartesian productS×S, ?nd µ (P,P ) = inf c(x,y)P(dx, dy), (1.1.1) c 1 2 wherethein?mumistakenoverallprobabilitymeasuresP onS×S having projections ?P = P,i=1,2. (1.1.2) i i 2 1. Introduction The Kantorovich–Rubinstein transshipment problem (KRP): GivenP 1 andP onS ?nd 2 ? µ (P,P ) = inf c(x,y)Q(dx, dy), (1.1.3) 1 2 c where the in?mum is taken over all ?nite measuresQ onS×S having the marginal di?erence ?Q??Q = P ?P ; (1.1.4) 1 2 1 2 that is,Q(A×S)?Q(S×A)=P (A)?P (A) for all Borel setsA?S.