Let $\mu$ be a finite Borel measure on $\mathbb{R}$.Let $E$ be a compact subset of $\mathbb{R}$. Suppose $E\subset \bigcup\limits_{k}J_k$, where $\{J_k\}$ is a countable collection of open intervals. Prove there exists a finite subcollection $\{J_{k_l}\}$ of $\{J_k\}$ consisting of disjoint intervals such that $$\mu(E\cap(\bigcup\limits_{l}J_{k_l}))\ge (\mu (E))/2.$$
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