Let \((X, \mathscr{A}, \mu)\) be a general measure space and \(1 \leqslant p \leqslant r \leqslant q \leqslant \infty\).

Prove that \(\mathcal{L}^{p}(\mu) \cap \mathcal{L}^{q}(\mu) \subset \mathcal{L}^{r}(\mu)\) by establishing the inequality

\[\|u\|_{r} \leqslant\|u\|_{p}^{\lambda} \cdot\|u\|_{q}^{1-\lambda} \quad \forall u \in \mathcal{L}^{p}(\mu) \cap \mathcal{L}^{q}(\mu),\]

with \(\lambda=\left(\frac{1}{r}-\frac{1}{q}ight) /\left(\frac{1}{p}-\frac{1}{q}ight)\).

[use Hölder's inequality.]