# Does infinite outer measure implies infinite inner measure?

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.

• This is related. The question isn't quite the same, but the answers provide the essential arguments. Sep 17, 2018 at 18:07
• A subset of the reals can have zero inner measure and infinite outer measure. In fact, a set with zero inner measure can be such that for each interval $I,$ the outer measure of its intersection with $I$ has outer measure equal to the length of $I.$ Seemingly even stronger (but I believe this will follow automatically from the interval property), a set with zero inner measure can "fill up the real line" so much that, for each set $E$ of positive (including infinite) measure, its intersection with $E$ will have outer measure equal to the measure of $E.$ Sep 17, 2018 at 18:39
• Regarding the claims I just made, see Is there a maximum to the amount of disjoint non-measurable subsets of the unit interval with full outer measure?. See also this 5 February 2006 sci.math post. Sep 17, 2018 at 18:41

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.