Show that there exists open set $O_n\subset [0,1]$ for all $n\in N$ such that $\overline{O_n}=[0,1]$ and $m^*(\cap_{n=1}^\infty O_n)=0$.

Question

Show that there exists open set $O_n\subset [0,1]$ for all $n\in N$ such that $\overline{O_n}=[0,1]$ and $m^*(\cap_{n=1}^\infty O_n)=0$.

My work-

I feel like there is NO such a set. Because, by he Baire category theorem, intersection of dense open sets is not empty. This follows, $m^*(\cap_{n=1}^\infty O_n)\neq0$.

Or else, $\cap_{n=1}^\infty O_n$ should be a countable set. Could you help me to find such a set. Hint also would be nice.

Here $m^*$ is the outer measure.

Answer 1

Let $(r_n)$ be an ennumeration of the rationals in $[0,1]$. For each $k$ let $I_{kn}$ be the interval $(r_n-\frac 1 {2^{n+k}},r_n+\frac 1 {2^{n+k}})$. Let $O_k=\bigcup_n I_{kn}$. Check that this sequence has the required properties.