This topic came out today during a discussion with a colleague. I realized that a counter-example to his claim could be constructed if there exits a subset $A \subset [0,1]$ such that $0 < \mu(A) < 1$ (where $\mu$ is Lebesgue measure) and $A$ is dense in $[0,1]$ (with respect to the standard topology on $\mathbb{R}$). However, I am not sure whether such a set exists. So the question is: do such sets exist? If so, is there an explicit construction, if not, how does one prove non-existence?