If $A,B \subset \mathbb{R} $ such that $m^*(A)=0 $ and $A \cup B$ is Lebesgue measurable. Show that $B$ is Lebesgue measurable.

Question

I was trying to do this by showing $B$ to be intersection or union of some measurable sets but wasn't able to do it.Any other way?

Answer 1

Note that $B = (A \cup B) - (A - B)$. Now, $A - B \subset A$ implies $m^*(A-B)=0$, so $A - B$ is measurable. Therefore, $B$ is measurable.

Answer 2

$A$ is automatically measurable because it has measure $0$

$$ B = (A \cup B)\setminus(A \setminus B)= (A \cup B) \cap (A \cap B^c)^c $$

You can make $B$ out of measurable things with union, intersection and compliment, so $B$ is measurable.