Generalized Hlder inequality. Iterate Hlder's inequality to derive the following generalization: [intleft|u_{1} cdot u_{2} cdots u_{N}ight| d

Generalized Hölder inequality. Iterate Hölder's inequality to derive the following generalization:

\[\int\left|u_{1} \cdot u_{2} \cdots u_{N}ight| d \mu \leqslant\left\|u_{1}ight\|_{p_{1}} \cdot\left\|u_{2}ight\|_{p_{2}} \cdots\|\| u_{N} \|_{p_{N}}\]

for all \(p_{n} \in(1, \infty)\) such that \(\sum_{n=1}^{N} p_{n}^{-1}=1\) and all measurable \(u_{n} \in \mathcal{M}(\mathscr{A})\).