Creating a Lebesgue measurable set with peculiar property. [duplicate]
Solution 1:
I assume you know how to construct a closed nowhere dense set of positive Lebesgue measure, i.e., a so-called "fat Cantor set". Clearly, a fat Cantor set can be constructed inside any given open interval.
Let $\{I_n:n\lt\omega\}$ be the set of all open intervals $\{q,r\}$ where $q,r$ are rational numbers and $0\le q\lt r\le1$. Recursively construct a sequence of pairwise disjoint fat Cantor sets $A_1,B_1,A_2,B_2,\dots,A_n,B_n,\dots$ with $A_n,B_n\subseteq I_n$. The set $E=\bigcup_{n=1}^\infty A_n$ has the desired properties.