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

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 $ξ\in L^β_G(Ω_T)$ with some $β>1$ the decomposition theorem holds and any $β>1$ 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

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