The following is an exercise from Bruckner's Real Analysis:
Let $C$ be a Cantor set in $[0, 1]$ of measure $α$ ($0 ≤ α<1$). Does there exist a sequence ${\{J_k}\}$ of intervals with $\sum_{k=1}^∞ m(J_k ) < ∞$ such that every point of the set $C$ lies in infinitely many of the intervals $J_k$?
My attempt: Let ${\{q_n}\}$ be an enumeration of $C \cap \mathbb{Q}$ and consider for each $n$ an interval of length $\dfrac{1}{2^n}$ symmetric around $q_n$. Consider minimum integer $j_1$ that some arbitrary but fixed point $x \in C$ belongs to it then that $I_{j_1}$ contains infinitely many $q_n$ take the next greater integer $j_2 \ge j_1$ such that it contains $x$, and repeat the process...
Am I right? If so, why a Cantor set with positive measure then if this argument holds for any set?
HINT: If $\alpha>0$ there is an $n_0\in\Bbb Z^+$ such that $\sum_{k\ge n_0}m(J_k)<\alpha$, so $C\setminus\bigcup_{k\ge n_0}J_k\ne\varnothing$.