Lemma 21.1.5.label Let $(X, \cm)$ and $(Y, \cn = \sigma(\ce))$ be measurable spaces, and $f: X \to Y$ such that $f^{-1}(\ce) \subset \cm$, then $f$ is $(\cm, \cn)$-measurable.
Proof. Let $\mathcal{F}= \bracs{E \in \ce| f^{-1}(E) \in \cm}$.$\square$
Lemma 21.1.5.label Let $(X, \cm)$ and $(Y, \cn = \sigma(\ce))$ be measurable spaces, and $f: X \to Y$ such that $f^{-1}(\ce) \subset \cm$, then $f$ is $(\cm, \cn)$-measurable.
Proof. Let $\mathcal{F}= \bracs{E \in \ce| f^{-1}(E) \in \cm}$.$\square$