$\sigma$-Algebra generated by a simple characteristic function

Question

Suppose that $X:\Omega \rightarrow \mathbb{R}$. Let $A \in \mathcal{B}$ be a Borel set.

Suppose that $X = \sum_i a_i\chi_{A_i}$ is a simple random variable defined on the probability space $(\Omega,\mathcal{F},\mathbb{P})$. Show that $A_i \in \sigma(X)$.

Answer 1

$$\begin{equation} \begin{split} \sigma(X) & = X^{-1}(\mathscr{B}(\mathbb{R})) \\ & = \sigma(\{A_1,A_2,...,A_n\}) \end{split} \end{equation}$$

Hence, $A_i \in \sigma(X)$