Duality for Borel measurable cost functions

Abstract

We consider the Monge-Kantorovich transport problem in an abstract measure theoretic setting. Our main result states that duality holds if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c colon upper X times upper Y right-arrow left-bracket 0 comma normal infinity right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>:</mml:mo> <mml:mi>X</mml:mi> <mml:mo> × </mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">c:X\times Y\to [0,\infty )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an arbitrary Borel measurable cost function on the product of Polish spaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X comma upper Y"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>Y</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">X,Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In the course of the proof we show how to relate a non-optimal transport plan to the optimal transport costs via a “subsidy” function and how to identify the dual optimizer. We also provide some examples showing the limitations of the duality relations.