Let $K : \mathbb R^n \to (0,\infty)$ be radially symmetric function such that $\|K\|_{L^1(\mathbb R^n)}=1$ and let $T: L^2(\mathbb R^n)\to L^2(\mathbb R^n)$ be defined as $u\mapsto T \star u$.
Suppose that $u\ge 0$ almost everywhere and $u>0$ on a set of positive measure, then prove that $T u>0$ almost everywhere.
My attempt: Let $E$ be the set of positive measure on which $u>0$. Positivity of $K$ implies $$(Tu)(x)=\int_{\mathbb R^n} K(x-y)u(y)\ dy\ge \int_E K(x-y)u(y)\ dy >0$$ and so $Tu>0$ almost everywhere.
Is my proof correct? If not, how to approach the problem?