Let \((X, \mathscr{A}, \mu)\) be a finite measure space and let \(1 \leqslant q

(i) Show that \(\|u\|_{q} \leqslant \mu(X)^{1 / q-1 / p}\|u\|_{p}\).

[use Hölder's inequality for \(u \cdot 1\).]

(ii) Conclude that \(\mathcal{L}^{p}(\mu) \subset \mathcal{L}^{q}(\mu)\) for all \(p \geqslant q \geqslant 1\) and that a Cauchy sequence in \(\mathcal{L}^{p}\) is also a Cauchy sequence in \(\mathcal{L}^{q}\).

(iii) Is this still true if the measure \(\mu\) is not finite?