Let $f$ vanishes on an open set containing 0. So there exists $l>0$ such that $f$ vanishes on $B(0,2l).$ So we can choose $g\in C_c^\infty (\mathbb{R}^n)$ (supported on $B(0,l)$) such that $f*g$ vanishes on an open set ( vanishes on $B(0,l)$).

My question: Is it remains true if we replace open set by set of positive Lebesgue measure i.e. if $f$ vanishes on an positive Lebesgue measure set around 0, then can we find a nonzero $g\in C_c^\infty (\mathbb{R}^n)$ such that $f*g$ vanishes on a set of positive Lebesgue measure?