counterexample for a theorem in measure theory

Question

In our class we have shown that if a set $F \subset \mathbb{R}$ is measurable than for every $\epsilon > 0$, $\exists$ an open set $O$, containing $F$, such that $\mu^*(O - F) \leq \epsilon$ My question is how can I have a counter example if the set is not measurable? I should prove that it does not exist an open set such that the last property $\mu^*(O - F) \leq \epsilon$ is not valid?

Answer 1

The starting theorem, without some restriction, is false as it depends entirely in the measure. By example if $\mu$ is the counting measure then for every nonempty open set $O\subset \mathbb{R}$ you have that $\mu^*(O\setminus \{0\})=\infty $.

P.S.: I'm assuming here that $\mu^*$ is induced from $\mu$ and the collection of open sets.