Definition 1

Let $X$ be a locally compact space, $\mu$ a measure on $X$. For every mapping $f$ of $X$ into a Banach space $F$, and for every number $p$ such that $1 \leq p < +\infty$, we denote by $N_p(f, \mu)$, or simply $N_p(f)$, the positive number $(\int^* |f|^p d|\mu|)^{1/p}$.

Note that the number $N_p(f)$ may be equal to $+\infty$.