# Rational translates of Fat Cantor set as positive but not full measure for each interval

I was wondering how whether this works: Choose $A=\cup_{r \in Q}r+F$ where F denotes the Fat Cantor Set.

To be precise, if μ denotes Lebesgue measure, how would one show that this A as a Borel set A⊂R such that $0<μ(A∩I)<μ(I)$ for every interval I in R?

A is not R by Baire Category theorem. But I haven't made much progress thereafter.