# How can I construct such a Borel subset? [duplicate]

This is a problem from the book "Real analysis for graduate students" : Let $m$ be a Lebesgue measure. Construct a Borel subset $A$ of $\mathbb{R}$ such that $0<m(A\cap I)<m(I)$ for every open interval $I$.

I looked up Folland's book for this problem, and there is a very similar question that ask for not whole $\mathbb{R}$, but $[0,1]$. There is a hint given there and it says like any sub-interval of $[0,1]$ contains a cantor-type set of positive measure. Yes, I understood the idea of constructing a generalized cantor set of positive measure, but I still couldn't see why it works even in $[0,1]$ case. If it works, then I guess I can construct same set for all intervals $[n,n+1]$, and then taking their union, it'll be also Lebesgue measurable and will satisfy the conditions. I'll greatly appreciate any help. Thanks!

• Oh, sorry, I've just realized that. Thanks. – vgmath May 27 '15 at 18:40