Consider $f$ and $g$ to be two functions such that $f$ is supported on the unit ball and $g$ is a function that vanishes in the unit ball (for points when $|x|<1$), and is non-zero for $|x|\geq 1$ in $\mathbb{R}^n.$ Assume further that $f,g$ are in $L^p(\mathbb{R}^n).$
Then if we mollify these two functions for instance as $f_\epsilon = f*\rho_\epsilon$ where $\rho_\epsilon(x)=\epsilon^{-n}\rho(x/\epsilon)$ and $\rho$ is the standard approximation of the identity. Then is it true that $f_\epsilon g_\epsilon \to fg=0$ in $L^p$?
Hint. If the functions $f$ and $g$ have disjoint compact support then their supports are a positive distance apart, and also $fg\equiv 0$. Can you verify that the mollifications of $f$ and $g$ also have this property, when $\epsilon$ is sufficiently small?