Definition 12.1.1 ($\mathcal{L}^{p}$ Spaces). Let $(X, \cm, \mu)$ be a measure space, $E$ be a normed vector space, $f: X \to E$ be strongly measurable, and $p \in [1, \infty)$, then $f$ is $p$-integrable if
\[\norm{f}_{L^p(X; E)}= \norm{f}_{L^p(\mu; E)}= \norm{f}_{L^p(X, \cm, \mu; E)}= \braks{\int \norm{f}_E^p d\mu}^{1/p}< \infty\]
The set $\mathcal{L}^{p}(X; E) = \mathcal{L}^{p}(\mu; E) = \mathcal{L}^{p}(X, \cm, \mu; E)$ is the space of all $p$-integrable functions on $X$.