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

The left-curtain martingale coupling in the presence of atoms