A Note on Duality Theorems in Mass Transportation

The duality theory of the Monge–Kantorovich transport problem is investigated in an abstract measure theoretic framework. Let $$(\mathcal {X},\mathcal {F},\mu )$$ ( X , F , μ ) and $$(\mathcal {Y},\mathcal {G},\nu )$$ ( Y , G , ν ) be any probability spaces and $$c:\mathcal {X}\times \mathcal {Y}\rightarrow \mathbb {R}$$ c : X × Y → R a measurable cost function such that $$f_1+g_1\le c\le f_2+g_2$$ f 1 + g 1 ≤ c ≤ f 2 + g 2 for some $$f_1,\,f_2\in L_1(\mu )$$ f 1 , f 2 ∈ L 1 ( μ ) and $$g_1,\,g_2\in L_1(\nu )$$ g 1 , g 2 ∈ L 1 ( ν ) . Define $$\alpha (c)=\inf _P\int c\,dP$$ α ( c ) = inf P ∫ c d P and $$\alpha ^*(c)=\sup _P\int c\,dP$$ α ∗ ( c ) = sup P ∫ c d P , where $$\inf $$ inf and $$\sup $$ sup are over the probabilities P on $$\mathcal {F}\otimes \mathcal {G}$$ F ⊗ G with marginals $$\mu $$ μ and $$\nu $$ ν . Some duality theorems for $$\alpha (c)$$ α ( c ) and $$\alpha ^*(c)$$ α ∗ ( c ) , not requiring $$\mu $$ μ or $$\nu $$ ν to be perfect, are proved. As an example, suppose $$\mathcal {X}$$ X and $$\mathcal {Y}$$ Y are metric spaces and $$\mu $$ μ is separable. Then, duality holds for $$\alpha (c)$$ α ( c ) (for $$\alpha ^*(c)$$ α ∗ ( c ) ) provided c is upper-semicontinuous (lower-semicontinuous). Moreover, duality holds for both $$\alpha (c)$$ α ( c ) and $$\alpha ^*(c)$$ α ∗ ( c ) if the maps $$x\mapsto c(x,y)$$ x ↦ c ( x , y ) and $$y\mapsto c(x,y)$$ y ↦ c ( x , y ) are continuous, or if c is bounded and $$x\mapsto c(x,y)$$ x ↦ c ( x , y ) is continuous. This improves the existing results in Ramachandran and Ruschendorf (Probab Theory Relat Fields 101:311–319, 1995) if c satisfies the quoted conditions and the cardinalities of $$\mathcal {X}$$ X and $$\mathcal {Y}$$ Y do not exceed the continuum.