Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 23I / a

For $f\in L^{\infty }({\mathbb{R}}^n)$ and $g\in L^1({\mathbb{R}}^n)$, define their convolution byIt is bounded becauseIf ${\tau }_{h}g(z)=g(z+h)$, thenContinuity of translation in $L^1$ makes the right-hand side tend to zero with $h$, uniformly in $x$. Thus the convolution of L infinity and L1 functions is bounded and continuous.

Now let $f=1_A$ and $g(x)=1_A(-x)$. ThenAt zero this equals $|A|>0$. By continuity it remains positive throughout some open neighbourhood $U$ of zero. Positivity at $z$ means that some $y$ satisfies $y\in A$ and $y-z\in A$, so $z\in A-A$. HenceThis is the Steinhaus theorem.

The conclusion also holds when $|A|=\infty$. Since ${\mathbb{R}}^n$ is the union of bounded balls, some measurable subset $A\cap B(0,R)$ has finite positive measure. Its difference set lies inside $A-A$ and already contains a neighbourhood of zero.