Highlevel Characterization of sets with lebesgue measure equal to 0 [closed]

I have a theory on why lebesgue measure is non zero for crazy sets like cantor sets which are not made of intervals.

Let $S$ be the set for which you want to find measure.

Define : For $x \in S$, $f(x) = \lim_{\epsilon \rightarrow 0} \frac{\mu \left(S\cap [x-\epsilon,x+\epsilon]\right)}{2\epsilon} > 0$ then that set will have non zero lebesgue measure is my hypothesis.If for all $x \in S$, $f(x) = 0$ then lebesgue measure of the set $S$ is $0$. If the limit diverges then the set is not measurable.

This is my theory. Can we prove this wrong ?

For example, all intervals and cantor set will satisfy this condition.