# Show that E and the closure of E have the same Jordan outer measure

1. Sep 22, 2016

### cragar

1. The problem statement, all variables and given/known data
Let E be a bounded set in $R^n$
Show that E and the closure of E have the same Jordan outer measure

2. Relevant equations
Jordan outer measure is defined as $m^* J(E)=inf(m(b))$
where $B \supset E$ B is elementary.
3. The attempt at a solution

If E and the closure of E are the same there is nothing to prove.
other case cl(E) contains limit points of E that are not in E.
Lets first look at the measure of E. BY the definition of Jordan outer measure, The closure of E will contain the inf(B) and E is a subset of B and the closure. Let B, be the closure of E, The inf(B) will be the measure of E.
Now this is where Im a little stuck. The Close of E, has all its limit points, so the inf(E) is part of E, its like a closed interval. There is no set B such that inf(m(B)). Unless i pick B to be the set such that
$B=E+ \epsilon$ so the inf of the measure of B is E.

2. Sep 27, 2016