Proving Measurable sets

Question

Here is the question: If $A \subseteq B$ are two measurable sets, then $B-A$ is also measurable, and $$m(B-A)=m(B)-m(A).$$ Here is my attempt: Using the definition of Lebesgue measurability we need to show for any $E \subseteq {R}^n$ we have $$m^*(E)=m*(E \cap(B-A))+m^*(E-(B-A))$$ which I think is complicated to show, so I think using the properties of measurable set is easier. So by using the property: If $E$ is measurable, then $R^n-E$ is also measurable. So $A^c$ is measurable, and hence $B\cap A^c$ is measurable by the properties of a measure sets. And, here I see $B \cap A^c= B-A$, so $m(B \cap A^c)= m(B-A).$ My confusing part is with showing that $m(B-A)=m(B)-m(A)$

Answer 1

Notice that $A$ and $B-A$ are disjoint (since $A\subset B)$. This implies $$m(B)=m((B-A)\cup A)=m(B-A)+m(A).$$