Optimal and better transport plans I

We consider the Monge-Kantorovick transport problem in a purely measure theoretic setting, i.e. without imposing continuity assumptions on the cost function. It is known that transport plans which are concentrated on c-monotone sets are optimal, provided the cost function c is either lower semi-continuous and finite, or continuous and possibly attain infinity. We show that this is true in a more general setting, in particular for merely Borel measurable cost functions which are finite almost everywhere on an open set. In a previous paper Schachermayer and Teichmann considered strongly c-monotone transport plans and proved that every strongly c-monotone transport plan is optimal. We establish necessary and sufficient conditions on c-monotone transport plans to be strongly c-monotone.