  Beiglb\"ock and Juillet ("On a problem of optimal transport under marginal
martingale constraints") introduced the left-curtain martingale coupling of
probability measures $\mu$ and $\nu$, and proved that, when the initial law
$\mu$ is continuous, it is supported by the graphs of two functions. We extend
the later result by constructing the generalised left-curtain martingale
coupling and show that for an arbitrary starting law $\mu$ it is characterised
by two appropriately defined lower and upper functions.
  As an application of this result we derive the model-independent upper bound
of an American put option. This extends recent results of Hobson and Norgilas
("Robust bounds for the American Put") on the atom-free case.

Probability