  We consider the Monge-Kantorovich 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 may possibly attain the value infty. We show that
this is true in a more general setting, in particular for merely Borel
measurable cost functions provided that {c=infty} is the union of a closed set
and a negligible 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 that transport plans are
strongly c-monotone if and only if they satisfy a "better" notion of optimality
called robust optimality.
