Find all the measurable functions in a space

Question

Let $A_1,...,A_n$ be a finite partition of $X$. Let $\mathcal{F}=\sigma(A_1,...,A_n)$, where $\sigma(A_1,...,A_n)$ is the sigma algebra generated by $A_1,...,A_n$. Find all the measurable functions in $(X, \mathcal{F})$. (Intuitively, the answer should be $f=\sum_{i=1}^n b_i I_{A_i}$, where $b_i\in \overline{\mathcal{R}}$, but I don't know how to prove. Thank you in advance.)

Answer 1

Because $A_1, \dots, A_n$ is a partition, $\cal F$ just comprises the finite unions of sub-collections of $A_1, \dots, A_n$. Since singleton subsets of $\overline{\cal R}$ are measurable, each set $f^{-1}[\{x\}]$ must be measurable (i.e., in $\cal F$) for every $x \in \overline{\cal R}$. This implies that $\mathrm{ran}(f)$ is finite, say, $\{x_1, \ldots, x_k\}$. Then $f = \sum_{i=1}^k x_iI_{B_i}$, where the $B_i$ are finite unions of sub-collections of $A_1, \dots, A_n$ and are pairwise disjoint. Expanding the $B_i$ out into their constituent $A_j$s gives you a sum of the desired form.