Let \(\mathscr{A}=\sigma(\mathscr{G})\) be a \(\sigma\)-algebra on \(X\), where \(\mathscr{G}=\left\{G_{i}: i \in \mathbb{N}ight\}\) is a countable generator. Let \(g:=\sum_{i=1}^{\infty} 2^{-i} \mathbb{1}_{G_{i}}\). Show that \(\sigma(g)=\mathscr{A}\).