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In this article, a sublinear expectation induced by $G$-expectation isintroduced, which is called $G$-evaluation for convenience. As an application,we prove that any $\xi\in L^\beta_G(\Omega_T)$ with some $\beta>1$ thedecomposition 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

Some properties on $G$-evaluation and its applications to $G$-martingale decomposition