# Prove combination of two sets contains an open ball

1. Dec 17, 2011

### aegis90

So this was an exam question that our professor handed out ( In class. I didn't get the question right)

Let E be a subset of R^n, n>= 2. Suppose that E measurable and m(E)>0. Prove that:

E+E = {x+y: x in E, y in E } contains an open ball.

(The text Zygmund that we used showed an example that E-E defined in similar sense contains an open interval centered at the origin, where E is a subset of R. Stein had another problem that asked to show that E+E contains an open interval.

I'm assuming that's where he got the problem, but I'm not sure that the same method works, since he gave a hint to prove that the convolution: chi(e)*chi(e) is continuous at the origin. )