First of all I'll say that I did see many questions in this site that try to answer a similar question like mine, but I didn't find them usefull enough.
As a part of a proof of mine I would like to claim:
$$\lim_{r\rightarrow 0^+} \frac{m(\bigcup_{i\in\mathbb N}E_i\bigcap(x-r,x+r))}{2r}\leq \sum_{i\in\mathbb N}\lim_{r\rightarrow 0^+}\frac{m(E_i\bigcap(x-r,x+r))}{2r}$$
($x\in\mathbb R$, $m$ is lebesgue measure, $r>0$, $E_i\subset \mathbb R$). Assume That each RHS limit exists.
How about trying $x=0$ and $E_i = (2^{-i-1},2^{-i}]$.