# Math Help - Lebesgue outer measure.

1. ## Lebesgue outer measure.

Let A Rn and let |A|e be the Lebesgue outer measure.
Suppose A, B Rn and d(A,B)>0. How do I prove that |A B|e=|A|e+|B|e?

2. ## Re: Lebesgue outer measure.

Hi, I would try this:

If $0 where $\delta (x,y)=\sqrt{\sum_{i=1}^n |x_i-y_i|^2}$

then there exists open set $G\subset\mathbb{R}^n$ such that :
1. $A\subset G$
2. $B\cap G=\emptyset$ .

Since every open set in $(\mathbb{R}^n, \delta)$ is Lebesgue measurable, there holds

$\forall S\subset\mathbb{R}^n \,:\,|S|_e=|S\cap G|_e + |S - G|_e$.

Rewriting this for $S=A\cup B$ and using properties 1,2 of set $G$ seems to be a good way to solve your problem.