Prove that if E is positive lebesgue measurabel set, then E − E and E + E contain non-empty open sets.

Question

let E + E = {x + y : x, y ∈ E}, and define E − E similarly. Show that if E is a measurable subset of R of positive Lebesgue measure then E − E and E + E contain non-empty open sets.

I have seen the solution by using Fubini's theorem but can't we solve it with out using fubini ? I like get some hints or complete solution by different method. Any hints and ideas are appreciated.

Answer 1

Here are a few hints:

Try to first prove this:

If $E$ is a measurable set and $m(E)> 0$, for any $\alpha < 1$, there is an open interval $I$ such that $$m(E \cap I) > \alpha m(I).$$

Once you have done this, use it to get the existence of $I$ for $\alpha > \frac{3}{4} $. It can be shown that $$(-\frac{1}{2}m(I), \frac{1}{2} m(I))\subset E - E.$$

This hint is out of Folland's Real Analysis book.