Given are two measurable spaces $(X, \mathbb A)$ and $(Y, \mathbb B)$. Assume we have a collection $\{M_n\} \subset A$ of mutually disjoint sets such that $X= \cup_{n=0}^{\infty} M_n$. In addition, let's suppose that $f_n : M_n \to Y $ ($n \in \mathbb N$) are given functions and let $f: X \to Y$ by setting $f(x) = f_n(x)$ whenever $x \in M_n$.
Why does it hold true that f is measurable from $(X, \mathbb A)$ to $(Y, \mathbb B)$ if and only if $f_n$ is measurable from $(M_n, \mathbb A|_{M_n}) $ to $(Y, \mathbb B)$ ? (where the $\sigma$-algebras are defined by $\mathbb A|_{M} := \{ A \cap M : A \in \mathbb A\} $)
Just use the set theoretic identities: $f^{-1} (B)=\cup_n f_n^{-1} (B)$ and $f_n^{-1} (B) =M_n \cap f^{-1} (B)$.