Is it necessarily true that the following intersection has a positive measure?

Question

suppose that a set $S$ in $\mathbb{R}$ is a measurable set with measure greater than zero, and let $\mathcal{O}$ be an open cover of $S$, consisting of disjoint open intervals whose existence we know. It is then necessarily true that there exists an open interval $I$ in $\mathcal{O}$ such that the measure of the intersection $S\cap I$ is non-zero? If so, how does one go about proving it? (Any hint would be appreciated.)

Thanks

Answer 1

If I understand correctly, you have $S \subset \mathcal{O} := \bigsqcup_n I_n$ (disjoint union). Since $S$ is measurable, $\mu(S)=\sum_n \mu(S \cap I_n)$.