For an integrable function $f$ we define $$F(x)=\int^l_{-l}f(x-t)dt.$$ What is the relation between the set $\mathcal{Z}_F,\mathcal{Z}_f?$ ($\mathcal{Z}_F$ is the zero set of $F$)
My work: If $f$ vanishes on the set $E$ then $F$ vanishes on the set $$\bigcap_{-l\leq t\leq l}t+E,$$ where $t+E=\{t+e:e\in E\}.$