If the union is measurable, so each set is measurable?

Question

Let $(X,\mathcal{X})$ be a ordered pair, consisting of a arbitrary set $X$ and a $\sigma$-algebra $\mathcal{X}$.

If $A \in \mathcal{X}$ and $A=A_1 \cup A_2 \cup A_3$, with $A_i \in X$ and $A_i \cap A_j=\emptyset$, is possible state that each $A_i \in \mathcal{X}$?

In other words, if the finite union of pairwise disjoint sets is measurable then is each set measurable?

Answer 1

The answer is no. If you pick any two disjoint non-measurable $A_1,A_2$ sets in $[0,1]$ and set $A_3=([0,1] \backslash A_1) \backslash A_2$ you get a counterexample.