  We provide a general Doob-Meyer decomposition for $g$-supermartingale
systems, which does not require any right-continuity on the system. In
particular, it generalizes the Doob-Meyer decomposition of Mertens (1972) for
classical supermartingales, as well as Peng's (1999) version for
right-continuous $g$-supermartingales. As examples of application, we prove an
optional decomposition theorem for $g$-supermartingale systems, and also obtain
a general version of the well-known dual formation for BSDEs with constraint on
the gains-process, using very simple arguments.
