Convolution of L infinity and L1 functions ($L^{\infty }\ast L^1\subset C_b$)

For $f\in L^{\infty }({\mathbb{R}}^n)$ and $g\in L^1({\mathbb{R}}^n)$,is bounded by $\Vert f{\Vert }_{\infty }\Vert g{\Vert }_1$. Continuity follows fromand continuity of translation in $L^1$.

If a Lebesgue-measurable set $A\subset {\mathbb{R}}^n$ has positive measure, then its difference set $A-A$ contains an open neighbourhood of zero. For finite measure, the convolution $1_A\ast 1_{-A}$ is continuous and positive at zero; the general case follows by taking a finite positive-measure subset.

If a measurable set $E\subseteq [0,L]$ has measure $m(E)=L/2+\alpha$, then for $|t|<2\alpha$ the sets $E$ and $E+t$ lie in an interval of length $L+|t|$, soThus $(-2\alpha ,2\alpha )\subseteq E-E$. This quantitative overlap argument is a bounded form of the Steinhaus theorem.