  The objective of this paper is to establish the decomposition theorem for
supermartingales under the $G$-framework. We first introduce a $g$-nonlinear
expectation via a kind of $G$-BSDE and the associated supermartingales. We have
shown that this kind of supermartingales have the decomposition similar to the
classical case. The main ideas are to apply the uniformly continuous property
of $S_G^\beta(0,T)$, the representation of the solution to $G$-BSDE and the
approximation method via penalization.
