  In this article, a sublinear expectation induced by $G$-expectation is
introduced, which is called $G$-evaluation for convenience. As an application,
we prove that any $\xi\in L^\beta_G(\Omega_T)$ with some $\beta>1$ the
decomposition theorem holds and any $\beta>1$ integrable symmetric
$G$-martingale can be represented as an It$\hat{o}'s$ integral w.r.t
$G$-Brownian motion. As a byproduct, we prove a regular property for
$G$-martingale: Any $G$-martingale $\{M_t\}$ has a quasi-continuous version
