I've been struggling with this exercise, and was hoping someone here might be able to point me in the right direction.
Question. Let $m^*$ denote the Lebesgue outer measure on $[0,1]$. Show that there exists $A \subseteq [0,1]$ s.t. for every interval $I \subseteq [0,1]$ we have
- $m^*(A \cap I)<m^*(I)$,
- $m^*(A^\complement \cap I)<m^*(I).$
EDIT: I guess I misunderstood the conditions, and this is indeed a duplicate. Apologies.