I am studying Bochner integral, reading K.Yosida "Functional Analysis 6th edition." However, I cannot understand the proof of Pettis' theorem, which is in p.131-132. From my perspective, this proof assumes the following proposition in its last sentence, but I cannot show it.
Proposition.
Let $(S,\mathcal{F}, \mu)$ be a measure space, and let $X$ be a Banach space. And let $f_n:S \to X$ is strongly measurable and $f_n(s) \to f(s)$ in $X$ for $a.e.s \in S$. Then, $f$ is strongly measurable.
Definition of simple and strongly measurable are as follows:
Definition.
Let $(S,\mathcal{F}, \mu)$ be a measure space, and let $X$ be a Banach space. $f:S \to X$ is called simple if there exist $x_1, \dots, x_N \in X$ and $X_1, \dots , X_N \in \mathcal{F}$ such that $f(s) = \sum_{n=1}^{N} \chi_{X_n}(s) x_n$ for all $s \in S$. $f: S \to X$ is called strongly measurable if there exists a sequence of simple functions $\{f_n\}_{n=1}^\infty$ such that $f_n(s) \to f(s)$ in $X$ for $a.e. s \in S$