Corollary 25.6.6.label Let $(X, \cm)$ be a measurable space, $(E, \norm{\cdot}_{E})$ be a separable normed vector space, and $f: X \to E$, then the following are equivalent:
- (1)
$f$ is $(\cm, \cb_{E})$-measurable.
- (2)
There exists simple functions $\seq{f_n}$ such that $\abs{f_n}\le \abs{f}$ for all $n \in \natp$, and $f_{n} \to f$ pointwise.
Proof. (1) $\Rightarrow$ (2): Let
then
- (AA1)
For each $y \in E$, $y \in \ol{\mathcal{A}(y)^o}$.
- (AA2)
$0 \in \bigcap_{y \in E}\mathcal{A}(y)$.
- (B)
For any fixed $y_{0} \in E \setminus \bracs{0}$,
\[\bracs{y \in E|y_0 \in \mathcal{A}(y)}= \bracs{y \in E|\norm{y_0}_E < \norm{y}_E}\cup \bracs{0}\in \cb_{E}\]and $\bracs{y \in E|0 \in \mathcal{A}(y)}= E$.
so $\mathcal{A}$ is a Borel measurable admissible approximant function.
By (2) of Corollary 25.6.5, there exists simple functions $\seq{f_n}$ such that $|f_{n}| \le |f|$ on $\bracs{f \ne 0}$ for all $n \in \natp$ and $f_{n} \to f$ pointwise. In which case, $|\one_{\bracs{f \ne 0}}f_{n}| \le |f|$ globally for all $n \in \natp$ and $\one_{\bracs{f \ne 0}}f_{n} \to f$ pointwise as $n \to \infty$.
(2) $\Rightarrow$ (1): By Proposition 25.5.4.$\square$
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