Definition 17.1.1 (Strongly Measurable Function). Let $(X, \cm)$ be a measurable space, $E$ be a normed vector space over $K \in \RC$, and $f: X \to E$, then the following are equivalent:
For each $\phi \in E^{*}$, $\phi \circ f$ is $(\cm, \cb_{K})$-measurable and $f(X) \subset E$ is separable.
$f$ is $(\cm, \cb_{E})$ measurable and $f(X) \subset E$ is separable.
There exists a sequence $\seq{f_n}\subset \Sigma(X, \cm; E)$ such that
For each $n \in \natp$, $\norm{f_n}_{E} \le \norm{f}_{E}$.
$\norm{f_n(x) - f(x)}_{E} \to 0$ pointwise as $n \to \infty$.
Proof. (1) $\Rightarrow$ (2): TODO
(2) $\Rightarrow$ (3): By Proposition 15.5.6.
(3) $\Rightarrow$ (1): For each $\phi \in E^{*}$, $\phi \circ f = \limv{n}\phi \circ f_{n}$ is measurable by Proposition 15.3.2. Since
and each $f_{n}$ is finitely-valued, $f(X)$ is separable.$\square$