Let $E$ and $F$ be the positive Lebesgue measure sets on $\mathbb{R}$, and let $e$ and $f$ be the points of density of $E$ and $F$, respectively. Then for any $\epsilon$, there exists a $\delta$ such that $$m([e-\delta, e+\delta] \setminus E)/ 2\delta < \epsilon,\text{ and } m([f-\delta, f+\delta] \setminus F)/ 2\delta < \epsilon.$$
Is the above statement true? I don't think it is. I guess. The $\delta$s cannot be the same. Can anyone tell me if it is true? If not, anyone can give me a counterexample?
The statement is true. By definition, if $e$ is a density point for $E$ then $$\lim_{\delta\to 0}\frac{m([e-\delta,e+\delta]\cap E)}{2\delta}=1$$ therefore $$\lim_{\delta\to 0}\frac{m([e-\delta,e+\delta]\backslash E)}{2\delta}=\lim_{\delta\to 0}\frac{m([e-\delta,e+\delta])-m([e-\delta,e+\delta]\cap E)}{2\delta}=0.$$ The same holds for $f$ and the conclusion follows from the definition of limit.