Let $X,Y$ be two standard Borel spaces. Let $\mu,\nu$ be probability measures on $X$ and $Y$, and let $c_1,c_2 : X\times Y \to [0,1]$ be two measurable (but not necessarily lower semi-continuous) maps. Assume that:
- for all $x\in X$, $c_1(x,\nu) = c_2(x,\nu)$ (where $c_i(x,\nu)$ denotes the pushforward of $\nu$ along the map $y \mapsto c_i(x,y)$);
- for all $y\in Y$, $c_1(\mu,y) = c_2(\mu,y)$.
Is it necessarily the case that the cost functions $c_1$ and $c_2$ induce the same optimal transport cost between $\mu$ and $\nu$? In other words, do we always have $$ \inf\left\{\int c_1(x,y)\,d\rho(x,y);\,\rho\in \Gamma(\mu,\nu)\right\} = \inf\left\{\int c_2(x,y)\,d\rho(x,y);\,\rho \in \Gamma(\mu,\nu)\right\},$$ where $\Gamma(\mu,\nu)$ denotes the set of all probability measures on $X\times Y$ with marginals $\mu$ and $\nu$?