Proposition 21.5.3.label Let $(X, \cm)$ be a measurable space, $Y$ be a metrisable topological space, and $f: X \to Y$ be a function, then the following are equivalent:
- (1)
$f$ is $(\cm, \cb_{Y})$-measurable.
- (2)
For each $\phi \in C(X; [0, 1])$, $\phi \circ f$ is $(\cm, \cb_{\real})$-measurable.
Proof. (2) $\Rightarrow$ (1): For each $U \subset X$ open, the function
\[d_{U^c}: X \to [0, 1] \quad x \mapsto d(x, U^{c}) \wedge 1\]
is continuous by Proposition 8.2.2. By Proposition 8.2.4, $\bracsn{d_{U^c} > 0}= U$. Thus $\bracs{f \in U}= \bracsn{d_{U^c} \circ f > 0}$.$\square$