Let \((X, \mathscr{A})\) and \(\left(X^{\prime}, \mathscr{A}^{\prime}ight)\) be measurable spaces and \(T: X ightarrow X^{\prime}\).

(i) Show that \(\mathbb{1}_{T^{-1}\left(A^{\prime}ight)}(x)=\mathbb{1}_{A^{\prime}} \circ T(x) \quad \forall x \in X\).

(ii) \(T\) is measurable if, and only if, \(\sigma(T) \subset \mathscr{A}\).

(iii) If \(T\) is measurable and \(u\) is a finite measure on \((X, \mathscr{A})\), then \(u \circ T^{-1}\) is a finite measure on \(\left(X^{\prime}, \mathscr{A}^{\prime}ight)\). Does this remain true for \(\sigma\)-finite measures?