Definition of Lebesgue Outer Measure: Given a set $E$ of $\mathbb R$, we define the Lebesgue Outer Measure of $E$ by, $$m^*(E) = \inf \left\{\sum_{n=1}^{+\infty} \ell(I_n): E \subset \bigcup_{n=1}^{+\infty}I_n \right\}$$ where $\ell(I_n)$ denotes the length of interval (bounded and nonempty interval).
Definition of measurable set: A set $E$ measurable if $$m^*(T) = m^*(T \cap E) + m^*(T \cap E^c)$$ for every subset of $T$ of $\mathbb R$.
My question: If $A_n$ is measurable for each $n \in \mathbb{N}$, is it true that $A_n^{\bullet}=A_n \backslash (A_1 \cup \cdots \cup A_{n-1})$, is measurable, for $n \geq 2$?
I proved that the finite union of measurable sets is measurable and that the measure of the union is the sum of the measures if they are pairwise disjoints. I don't know if that helps anything.