Let $\lbrace I_1,\ldots I_k \rbrace$ be a collection of bounded intervals. Choose $I_1$ to be of the largest. Denote $T=\lbrace i\in \lbrace 1,\ldots ,k\rbrace \mid (I_1 \cap I_i)\not= \emptyset\rbrace $.
I want to know why is it the case that if $T=\lbrace1,\ldots ,k\rbrace$ then $\mu(I_1)\ge\frac{1}{3}\mu(\cup_{i=1} ^k I_i)$.