$f,g \in L^1 (\Omega)$, $\Omega\subset\mathbb{R}^n$ is a Lipschitz-domain.
Prove that $$(\forall\phi\in C^{\infty}_{C}(\Omega))\Big(\int_{\Omega}^{}f*\phi=\int_{\Omega}^{}g*\phi\Big)\Rightarrow f=g $$
where $\phi\in C^\infty_C(\Omega) \Rightarrow f\in C^\infty(\Omega)$ and $ supp f \subset\Omega$ is compact.
I'm working on a project, and I'm stuck on this proof here,so if anyone could help me I would be most grateful