Let $A$ be any subset of $\mathbb R$ and let $I$ be any compact interval of $\mathbb R$ containing $A$
Then I need to prove that $$L(I)=m(I \setminus A)+m(A)$$ where $L(I)$ denotes the length of $I$ and $m(A)$ denotes the outer measure of $A$
All I know regarding this problem is, if $F$ is collection of subsets of $\mathbb R$ such that $A$ is in $F$ if $m(A\cup B)=m(A)+m(B)$, where $B$ is any subset of $\mathbb R$ such that $A\cap B = \varnothing$, then this collection contains all intervals.