Show that, for the squared-error loss, the approximation error (ellleft(g^{mathscr{C}} ight)-ellleft(g^{*} ight)) in (2.16), (begin{array}{llllll}text { is } quad text { equal } & text

Show that, for the squared-error loss, the approximation error \(\ell\left(g^{\mathscr{C}}\right)-\ell\left(g^{*}\right)\) in (2.16), \(\begin{array}{llllll}\text { is } \quad \text { equal } & \text { to } & \mathbb{E}\left(g^{\mathscr{G}}(\boldsymbol{X})-g^{*}(\boldsymbol{X})\right)^{2} & \text { [Hint: } & \text { expand } \\ \left.\ell\left(g^{\mathscr{G}}\right)=\mathbb{E}\left(Y-g^{*}(\boldsymbol{X})+g^{*}(\boldsymbol{X}) g^{\mathscr{G}}(\boldsymbol{X})\right)^{2} .\right] & & \end{array}\)