Let \(E \in \mathscr{B}(\mathbb{R}), Q: E ightarrow \mathbb{R}, Q(x)=x^{2}\), and \(\lambda_{E}:=\lambda(E \cap \cdot)\) (Lebesgue measure).

(i) Show that \(Q\) is \(\mathscr{B}(E) / \mathscr{B}(\mathbb{R})\)-measurable.

(ii) Find \(u \circ Q^{-1}\) for \(E=[0,1], u=\lambda_{E}\) and \(E=[-1,1], u=\frac{1}{2} \lambda_{E}\).