Let \((X, \mathscr{A}, \mu)\) be a measure space and \(u, v \in \mathcal{L}^{p}(\mu)\).

(i) Find conditions which guarantee that \(u v, u+v\) and \(\alpha u, \alpha \in \mathbb{R}\) are in \(\mathcal{L}^{p}(\mu)\).

(ii) Show that \(\mathcal{L}^{1}(\mu)\) and \(\mathcal{L}^{2}(\mu)\) are, in general, no algebras.

(iii) Show that the lower triangle inequality holds:

\[\left|\|u\|_{p}-\|v\|_{p}ight| \leqslant\|u-v\|_{p}\]