- #1

cragar

- 2,552

- 3

## Homework Statement

Let E be a bounded set in [itex] R^n [/itex]

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

## Homework Equations

Jordan outer measure is defined as [itex] m^* J(E)=inf(m(b)) [/itex]

where [itex] B \supset E [/itex] B is elementary.

3. The Attempt at a Solution [/B]

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.

Let's 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 I am 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

[itex] B=E+ \epsilon [/itex] so the inf of the measure of B is E.