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We prove that $c$-cyclically monotone transport plans $\pi$ optimize theMonge-Kantorovich transportation problem under an additional measurabilitycondition. This measurability condition is always satisfied for finitelyvalued, lower semi-continuous cost functions. In particular, this yields apositive answer to Problem 2.25 in C. Villani's book. We emphasize that we donot need any regularity conditions as were imposed in the previous literature.

Characterization of optimal Transport Plans for the Monge-Kantorovich-Problem