We provide a general Doob-Meyer decomposition for $g$-supermartingale systems, which does not require any right-continuity on the system, nor that the filtration is quasi left-continuous. In particular, it generalizes the Doob-Meyer decomposition of Mertens  for classical supermartingales, as well as Peng’s  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 formulation for BSDEs with constraint on the gains-process, using very simple arguments.
"A general Doob-Meyer-Mertens decomposition for g-supermartingale systems." Electron. J. Probab. 21 1 - 21, 2016. https://doi.org/10.1214/16-EJP4527