I need your assistance in proving the following lemma.
Let $E\subset [0,2]$ be a set of positive Lebesgue measure. Then $A = \{x \in [0,1] : 2x \in E\}$ has positive measure.
A sketch of my idea on its proof.
To prove the lemma, it suffices to show that its contrapositive is true.
Suppose $m(A) = 0$. Let $\epsilon >0$. There exists a collection of open intervals $\{I_k\}_k^\infty$ such that $A \subset\cup_k^\infty I_k$ and $\sum_k^\infty m(I_k)< \epsilon$.
My challenge is on how to extend this cover of $D$ to a cover of the set $E$.
I was thinking about setting $I_k^\prime = I_k \cup I_k+a$ for some constant $a$. My concern is on how to choose $a$ so that ${I_k^\prime}$ will be a cover of $E$ by open intervals.
Can anybody help me with a suitable $a$ or an alternative way to prove the lemma?