Corollary 25.6.5.label Let $(X, \cm)$ be a measurable space, $Y$ be a separable metric space, and $\mathcal{A}: Y \to 2^{Y}$ be a Borel measurable admissible approximant function, then for any $f: X \to Y$, the following are equivalent:
- (1)
$f$ is $(\cm, \cb_{Y})$-measurable.
- (2)
For any dense subset $\seq{y_n}\subset Y$ with $y_{1} \in \bigcap_{y \in Y}\mathcal{A}(y)$, there exists a sequence $\seq{f_n}$ of $(\cm, \cb_{Y})$-measurable simple functions such that
- (i)
For each $x \in X$ and $N \in \natp$,
\[f_{N}(x) \in \mathcal{A}(f(x)) \cap \bracsn{y_n|1 \le n \le N}\] - (ii)
$f_{n} \to f$ pointwise as $n \to \infty$.
- (3)
There exists a sequence $\seq{f_n}$ of $(\cm, \cb_{Y})$-measurable simple functions such that $f_{n} \to f$ pointwise.
Proof. (1) $\Rightarrow$ (2): Let $\seq{y_n}\subset Y$ be a dense subset with $y_{1} \in \bigcap_{y \in Y}\mathcal{A}(y)$. By Lemma 25.6.4, there exists $\seq{I_n}\subset Y^{Y}$ such that:
- (1)
$\seq{I_n}$ is an $\mathcal{A}$-admissible approximation of the identity.
- (2)
For each $N \in \natp$, $I_{N}$ is Borel measurable with $I_{N}(Y) \subset \bracsn{y_n|1 \le n \le N}$.
For each $n \in \natp$, let $f_{n} = I_{N} \circ f_{n}$, then:
- (i)
For each $x \in X$ and $N \in \natp$,
\[f_{N}(x) = I_{N}(f(x)) \in \mathcal{A}(f(x)) \cap \bracsn{y_n|1 \le n \le N}\] - (ii)
Since $I_{n} \to \text{Id}$ pointwise as $n \to \infty$, $f_{n} \to f$ pointwise as $n \to \infty$.
(3) $\Rightarrow$ (1): By Proposition 25.5.4.$\square$
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