Let A R^{n} and let |A|_{e} be the Lebesgue outer measure.
Suppose A, B R^{n} and d(A,B)>0. How do I prove that |A B|_{e}=|A|_{e}+|B|_{e}?
Having thought about this problem for days, I still can't solve it. Any suggestion?
Hi, I would try this:
If where
then there exists open set such that :
1.
2. .
Since every open set in is Lebesgue measurable, there holds
.
Rewriting this for and using properties 1,2 of set seems to be a good way to solve your problem.