Prove or give a counter example: For every open set $U$ of $\mathbb{R}$, $m^*(\bar{U} - U ) = 0.$

My first impression was that it was true, since if $U$ is an open set in $\mathbb{R}$, then it can be expressed as the disjoint union of a countable collection of open intervals. We also know that the outer measure of an interval is just its length. So, then I get stuck by wondering if $\bar{U} - U$ is always at most countable, which at the moment I cannot think of a case where it's not true. For if this is true, then the outer measure of $\bar{U}-U$ would be zero and I would be done.

What am I missing here?


Hint: http://en.wikipedia.org/wiki/Smith%E2%80%93Volterra%E2%80%93Cantor_set

It is a closed set whose boundary has positive measure. Hence the complement...

  • $\begingroup$ The complement obviously is an open set, but I guess I'm not seeing why the boundary of the complement still has positive measure. As a disclaimer, I'm using Royden/Fitzpatrick, 4th Edition. We are up to Section 2.2. $\endgroup$ – user70482 Apr 21 '13 at 22:13
  • $\begingroup$ What's the relationship between the boundary of a set and the boundary of it's complement? $\endgroup$ – Zach L. Apr 21 '13 at 22:14
  • $\begingroup$ That the boundary of a set is the intersection of its closure and the closure of its complement? So the measure of the boundary of a set has the same measure of the boundary of its closure? $\endgroup$ – user70482 Apr 21 '13 at 22:22
  • $\begingroup$ The set $C$ described above is closed and the measure of its boundary is positive. What is the boundary of $\mathbb{R} \setminus C$ in terms of the boundary of $C$? $\endgroup$ – Zach L. Apr 21 '13 at 22:27
  • 1
    $\begingroup$ @Archie: You say that the boundary of a set $C$ is just $$\overline C\cap\overline{\Bbb R\setminus C}.$$ Since $\Bbb R\setminus(\Bbb R\setminus C)=C$ and since $A\cap B=B\cap A$, this means that a set and its complement have the same boundary. $\endgroup$ – Cameron Buie Apr 21 '13 at 22:33

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