Proposition 23.1.2.label Let $(X, \cm)$ be a measurable space, $E$ be a normed vector space over $K \in \RC$, then:
- (1)
For any strongly measurable functions $f, g: X \to E$ and $\lambda \in K$, $\lambda f + g$ is strongly measurable.
- (2)
For any strongly measurable functions $\bracs{f_n: X \to E|n \in \natp}$ and $f: X \to E$, if $f_{n} \to f$ strongly pointwise, then $f$ is strongly measurable.
Proof. (1): Since $x \mapsto \lambda$ is continuous, $\lambda f$ is strongly measurable by (2) of Definition 23.1.1.
By (3) of Definition 23.1.1, there exists $\seq{f_n}, \seq{g_n}\subset \Sigma(X, \cm; E)$ such that $f_{n} \to f$ and $g_{n} \to g$ strongly pointwise. In which case, $\seq{f_n + g_n}\subset \Sigma(X, \cm; E)$ and $f_{n} + g_{n} \to f + g$ strongly pointwise. Therefore $f + g$ is also strongly measurable.
(2): By Proposition 21.5.4, $f$ is $(\cm, \cb_{E})$-measurable. Since $f(X) \subset \overline{\bigcup_{n \in \natp}}f_{n}(X)$, $f(X)$ is also separable, so $f$ is strongly measurable by (1) of Definition 23.1.1.$\square$