Suppose that $\displaystyle p_{j}\in[1, \infty]$ for $\displaystyle j = 1,...,m$ and let $\displaystyle 1/r = \sum_{j=1}^{m}1/p_{j}$.

For $\displaystyle f_{j}\in{L_{p_{j}}\left(X, \mu, \mathbb{K}\right)$, show that $\displaystyle \prod_{j=1}^{m}\in{L_{r}\left(X, \mu, \mathbb{K}\right)$ and that:

$\displaystyle \displaystyle \biggl\|\prod_{j=1}^{m} f_{j} \biggl\|_{r} \le \prod_{j=1}^{m} \|{f_j}\|_{p_j}$