Showing that if A$\subset$E$\subset$B and lebesgue measures of A and B are equal and finite, then E is lebesgue measurable.

Question

I am working on the following question:

Suppose that A ⊂ E ⊂ B, where A and B are λ-measurable sets of finite measure. Prove that if λ(A) = λ(B), then E is λ-measurable.

I am working with the Lebesgue measure on $\mathbb R$.

I have currently:

If A, B are λ-measurable sets of finite measure then they must also be λ*-measurable sets of finite measure, where λ* denotes the Lebesgue outer measure on $\mathbb R$.

Meaning they must satisfy λ*(S)=λ*(S$\cap$A)+λ*(S$\cap$$A^c$) and λ*(S)=λ*(S$\cap$B)+λ*(S$\cap$$B^c$).

I also must have that: A$\subset$E, so λ*(A)$\le$λ*(E) and E$\subset$B so λ*(E)$\le$λ*(B) as a consequence of outer measure. But I am unsure of how to progress with this problem.

Any help would be greatly appreciated.

Answer 1

$E\setminus A\subseteq B\setminus A,$ and $B\setminus A$ is Lebesgue measurable with measure $0.$ It follows that $E\setminus A$ is also measurable (with measure $0),$ so $E=A\cup (E\setminus A)$ is measurable, being the union of two measurable sets.