In this paper, we introduce the weak G $\mathrm{<span\; style="font-family:\; 돋움;\; font-size:\; 11pt;">G</span>}$ -expansivity which is a generalization of both expansivity and G $\mathrm{<span\; style="font-family:\; 돋움;\; font-size:\; 11pt;">G</span>}$ -expansivity. Also, we define G $\mathrm{<span\; style="font-family:\; 돋움;\; font-size:\; 11pt;">G</span>}$ -stable and G $\mathrm{<span\; style="font-family:\; 돋움;\; font-size:\; 11pt;">G</span>}$ -unstable sets of a homeomorphism on a metric G $\mathrm{<span\; style="font-family:\; 돋움;\; font-size:\; 11pt;">G</span>}$ -space X $\mathrm{<span\; style="font-family:\; 돋움;\; font-size:\; 11pt;">X</span>}$ and investigate properties of them. Finally, we consider the decomposition theorem on G $\mathrm{<span\; style="font-family:\; 돋움;\; font-size:\; 11pt;">G</span>}$ -spaces.