Let \((X, \mathscr{A}, \mu)\) be a \(\sigma\)-finite measure space and assume that \(u=f \mu\) for a positive measurable function \(f\).

(i) Show that \(u\) is a finite measure if, and only if, \(f \in \mathcal{L}^{1}(\mu)\).

(ii) Show that \(u\) is a \(\sigma\)-finite measure if, and only if, \(\mu\{f=\infty\}=0\).