Let $E \subseteq \mathbb{R}$. Show that the following are equivalent:
(i) $E$ is Lebesgue measurable ;
(ii) $|I| \geq m(I \cap E)+m(I \backslash E)$ forall interval $I \subset \mathbb{R}$ of finite lenght;
(iii) $E \cap[n, n+1)$ is measurable forall $n \in \mathbb{Z}$;
(iv) $m([n, n+1) \cap E)+m([n, n+1) \backslash E)=1$ forall $n \in \mathbb{Z}$.
There is only one thing I have to show: (iv) implies (ii), because I have shown that i) and ii) are equivalent. I was thinking about that if (iv) is true for all $n$ if I replace $[n, n+1)$ with $I=\bigcup_{n \in \mathbb{Z}}([n, n+1)\cap I)$ and I write 1 as $[n, n+1)$ is still right and I have the result? Are there other ways?