  We give an injective martingale coupling; in particular, given measures $\mu$
and $\nu$ in convex order on $\mathbb{R}$ such that $\nu$ is continuous, we
construct a martingale coupling $\pi$ of the two measures with disintegration
$\pi(dx,dy) = \pi_x(dy) \mu(dx)$ such that $\#\{ x : y \in \textrm{supp}( \pi_x
) \} = 1$ for $\nu$ almost all $y$. Equivalently we find a martingale coupling
$\pi$ such that $\pi$ has disintegration $\pi(dx,dy) = \nu(dy)
\delta_{\theta(y)}(dx)$ for some function $\theta$.
