Understanding a part of the proof that a linear combination of 2 singular measures is a singular measure.

Question

Here is the problem and its solution:

My question is:

I do not understand, in the solution, why $$ X = (A_{1} \cap A_{2}) \cup (B_{1} \cup B_{2}),$$ Could anyone explain this for me please?

Answer 1

Suppose $x \notin B_1 \cup B_2$. Then $x \notin B_1$ which implies $x \in A_1$ because $X =A_1 \cup B_1$. Similarly, $x \notin B_2$ which implies $x \in A_2$ because $X =A_2 \cup B_2$. So $x \in A_1 \cap A_2$. Hence any point $x$ either belongs to $B_1 \cup B_2$ or to $A_1\cap A_2$.