Uniqueness of the transport plan satisfying the super-martingale constraint

Question

Let $\mu,\nu$ be probability measures on $\mathbb R$ such that

$$\int_{\mathbb R}|x| \mu(d x) + \int_{\mathbb R}|x| \nu(d x)<\infty.$$

$\mu\le_{cd} \nu$ is said to hold if

$$\int_{\mathbb R}(x-y)^+\mu(d y) \le \int_{\mathbb R}(x-y)^+ \nu(dy),\quad \forall x\in\mathbb R.$$

Denote by $S(\mu,\nu)$ the collection of probability measures $\pi$ on $\mathbb R^2$ of marginal distributions $\mu,\nu$ so that
$$\int_{\mathbb R}y\pi_x(d y)\le x,\quad \mbox{for } \mu-\mbox{almost every } x\in\mathbb R,$$ where $(\pi_x)_{x\in \mathbb R}$ denotes the regular conditional disintegration of $\pi$ with respect to the first marginal $\mu$, i.e. $\pi(d x,d y)=\mu(d x)\pi_x(d y)$. It is known that $S(\mu,\nu)\neq \emptyset$ if and only if $\mu\le_{cd} \nu$. My question is as follows :

$$\mbox{If } S(\mu,\nu) \mbox{ contains only one element, do we have } \mu=\nu?$$

Answer 1

The answer is non due to the very simple example. Put

$$\mu=\delta_0 \quad\mbox{and}\quad \nu=p\delta_{1}+(1-p)\delta_{-1}$$

where $p\in (0,1)$. A straightforward verification yields $\mu\le_{cd}\nu$ if $p\le 1/2$. Clearly, for every $0<p\le 1/2$, $S(\mu,\nu)$ only contains one element while $\mu\neq \nu$.