Lebesgue measure - How to prove A + A contains an open ball?

Question

I've been trying to prove the following, but haven't got there yet, maybe someone can give some hints.

Let A be a subset of R^n with positive Lebesgue measure. Then, the set A + A = {a1 + a2 | a1 and a2 are in A} contains an open ball. I could prove this for the set A - A, but not to A + A

Answer 1

First assume that m(A) is finite also. Define
f(x)=int_A chi_A (x-y) dy.
The f is continuous and positive and int f(x)dx=m(A)^2 >0,
so f is positive on some ball. But if f(x)>0, you have y in A with x-y also in A, so x is in A+A.