$m(A) + m(B) = m(A \cup B) + m(A \cap B)$?

Question

Does $m(A) + m(B) = m(A \cup B) + m(A \cap B)$ extend to unbounded sets $A,B$. Does unboundedness change anything?

Answer 1

Just write $$A=(A\setminus B) \cup (A\cap B)$$ and $$B=(B\setminus A)\cup(A\cap B)$$ and note these decompositions are disjoint. Then $$A\cup B = (A\setminus B)\cup(B\setminus A)\cup(A\cap B)$$ is also a disjoint decomposistion.

So: $$m(A)+m(B) = m(A\setminus B) + m(B\setminus A) + 2m(A\cap B) = m(A\cup B)+m(A\cap B)$$

So the thing you need to know is that if $X,Y$ are measurable, then $X\setminus Y$ and $X\cap Y$ are measurable, too.