CONTENTS Introduction...................................................................................5 §1. Preliminaries...........................................................................7 §2. Definitions and a theorem of Diestel, Faires and Huff.............9 §3. Examples...............................................................................13 §4. Some special classes of Boolean algebras ..........................19 §5. The Grothendieck property...................................................22 §6. The Orlicz-Pettis property ....................................................27 References.................................................................................32
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The duality theory for the Monge-Kantorovich transport problem is analyzed in a general setting. The spaces X,Y are assumed to be Polish and equipped with Borel probability measures μ and ν. The transport cost function c: X × Y → [0,∞] is assumed to be Borel. Our main result states that in this setting there is no duality gap provided the optimal transport problem is formulated in a suitably relaxed way. The relaxed transport problem is defined as the limiting cost of the partial transport of masses 1 - ε from (X,μ) to (Y,ν) as ε > 0 tends to zero. The classical duality theorems of H. Kellerer, where c is lower semicontinuous or uniformly bounded, quickly follow from these general results.
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