Let $A \subset E\subset B$ , where $A$ and $B$ are subsets of $\mathbb{R}$ of finite Lebesgue measure and $E$ is a subset of $\mathbb{R}$, then $E$ is also Lebesgue measurable if $\mu(A)=\mu(B)$ .
My attempt:- since $A$ is measurable then for any $C \subset R$ we have :- $$ \mu(C) \ge \mu ( C\cap A) + \mu(C \cap A')$$ $$\ge \mu(C\cap E)+\mu( C\cap B')$$ $$\ge \mu(C\cap E)+ \mu (C\cap E')$$ Hence $E$ is measurable , my attempt based on this :-
if $\mu(A)=\mu(B)$ then $\mu(A\cap E)=\mu(B\cap E)$ so is my attempt is right and did what I used is correct ?and if there is different way to solve this problem ?