Suppose that \(E\) and \(F\) are Lebesgue measurable sets of \(\mathbb{R}\), and their Lebesgue measures \(m(E) > 0\), \(m(F) > 0\). Prove that
\[E + F = \{x + y : x \in E, y \in F\}\]
contains a nonempty open interval.
Solution
Choose \(R>0\) so large that \[ E_0:=E\cap[-R,R], \qquad F_0:=F\cap[-R,R] \] both have positive measure. Then \(E_0,F_0\) have finite positive measure and \[ E_0+F_0\subseteq E+F. \]
Define \[ G(x):=(\mathbf1_{E_0}*\mathbf1_{F_0})(x) =\int_{\mathbb R}\mathbf1_{E_0}(x-y)\mathbf1_{F_0}(y)\,dy. \] Since \(\mathbf1_{E_0}\in L^1(\mathbb R)\) and translations are continuous in \(L^1\), \[ \begin{aligned} |G(x+h)-G(x)| &\le \int_{\mathbb R} |\mathbf1_{E_0}(x+h-y)-\mathbf1_{E_0}(x-y)| \mathbf1_{F_0}(y)\,dy\\ &\le \|\tau_h\mathbf1_{E_0}-\mathbf1_{E_0}\|_1 \longrightarrow0. \end{aligned} \] Thus \(G\) is continuous.
Moreover, Tonelli’s theorem gives \[ \int_{\mathbb R}G(x)\,dx =m(E_0)m(F_0)>0. \] Hence \(G(x_0)>0\) for some \(x_0\). By continuity there is \(\delta>0\) such that \[ G(x)>0 \qquad\text{whenever }|x-x_0|<\delta. \]
If \(G(x)>0\), then the set of \(y\) for which \[ y\in F_0, \qquad x-y\in E_0 \] has positive measure, hence is nonempty. Therefore \(x\in E_0+F_0\).
Consequently \[ (x_0-\delta,x_0+\delta) \subseteq E_0+F_0 \subseteq E+F. \] Thus \(E+F\) contains a nonempty open interval.