Let \((X, \mathscr{A}, \mu)\) be a finite measure space. Show that

(i) \(u \in \mathcal{L}^{1}(\mu) \Longleftrightarrow u \in \mathcal{M}(\mathscr{A})\) and \(\sum_{n=0}^{\infty} \mu\{|u| \geqslant n\}<\infty\);

(ii) \(\sum_{n=1}^{\infty} \mu\{|u| \geqslant n\} \leqslant \int|u| d \mu \leqslant \sum_{n=0}^{\infty} \mu\{|u| \geqslant n\}\).

(iii) The lower estimate in (ii) holds in an arbitrary measure space.