Does infinite outer measure implies infinite inner measure?

Question

Just came over this question in my mind: if I know a set has an infinite outer measure, does it mean it also has to have an infinite inner measure.

Answer 1

There exists $A\subset \Bbb R$ such that for any uncountable closed $C\subset \Bbb R$ we have $C\cap A\ne \emptyset \ne C \cap (\Bbb R\setminus A)$.

Any closed subset of $A$ is countable so $A$ has inner measure $0.$

Any closed subset of $\Bbb R\setminus A$ is also countable. So if $U$ is open and $U\supset A$ then $\Bbb R\setminus U$ is a closed subset of $\Bbb R\setminus A,$ so $\Bbb R\setminus U$ is countable, so $U$ has infinite measure.