Suppose $X_i$ are iid exponentials and $X(A) = \sum_{n=1}^\infty \mathbb{1}_{X_1 + \cdots+X_n \in A}$ for Borel set $A$. I have shown that $X$ is a random measure, and that $X(A) \sim \operatorname{Poisson}(|A|)$ where $|A|$ is the Lebesgue measure of $A$. I have also shown that for $A_1, \ldots, A_k$ disjoint intervals, $X(A_1), \ldots, X(A_k)$ are independent. But I need to show this for general Borel sets $A_i$.
We can extend to disjoint open sets by simply writing each open set as a union of disjoint open sets. But now what? How do we extend to all Borel sets?