# Is there a $G_\delta$ set with Positive Measure and Empty Interior?

It is like in the title. $G_\delta\subset\mathbb{R}^n$ with Lebesgue measure.

Thanks for any help

-
It could even be a closed set in $[0,1]$. – GEdgar Oct 26 '12 at 21:46

Yes: any fat Cantor set in $\Bbb R$ is an example, since all closed sets in $\Bbb R$ are $G_\delta$’s.

-
Thank you Brian – Tomás Oct 26 '12 at 21:49
@Tomás: You’re welcome. – Brian M. Scott Oct 26 '12 at 21:49

Yes. $\mathbb R\setminus \mathbb Q$.

-
Aw, that one’s boring! :-) +1 – Brian M. Scott Oct 26 '12 at 21:48
Thanks you kahen – Tomás Oct 26 '12 at 21:53
His example make me feel like a dumb @BrianM.Scott haha, but i like you better, because it is a nowhere dense set. – Tomás Oct 26 '12 at 22:17

Yes.

For $n\in\mathbb N$ let $F_n = \{x\in(0,1)|\;\exists m\in\mathbb Z:x=\frac{m}{n}\}$ and $U_n=(0,1)\setminus F_n$. The sets $U_n$ are open and have measure $1$, but the intersection $A=\bigcap_{n=1}^{\infty}U_n$ is simply the set of all irrational numbers in $(0,1)$. It is thus a $G_\delta$ set with measure $1$ and empty interior.

-