A question regarding $L^1 $and $C^\infty_C(\Omega)$ spaces $f,g \in L^1 (\Omega)$, $\Omega\subset\mathbb{R}^n$ 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 
 A: If $\Omega$ is not all $\mathbb{R}^{n}$, the function $$\int_{\Omega}f(x-y)\phi(y)dy$$
is not well defined for all $x$, so im gonna assume that $\Omega=\mathbb{R}^{n}$.
Let $F(x,y)=f(x-y)\phi(y)$. Note that \begin{eqnarray}
 \int_{\mathbb{R}^{n}}|F(x,y)|dx &=& \int_{\mathbb{R}^{n}}|f(x-y)|\phi(y)|dx      \nonumber \\
   &=& |\phi(y)|\|f\|_{L^{1}(\mathbb{R}^{n})} 
\end{eqnarray}
Hence, $$\int_{\mathbb{R}^{n}} \int_{\mathbb{R}^{n}}|F(x,y)|dxdy=\|\phi\|_{L^{1}(\mathbb{R}^{n})}\|f\|_{L^{1}(\mathbb{R}^{n})} 
$$
By Tonelli's theorem (see Brezis - Functional Analyis, Sobolev Spaces and PDE, page 91) $F\in L^{1}(\mathbb{R}^{n}\times\mathbb{R}^{n})$. 
Then by Fubini's theorem (see Brezis - Functional Analyis, Sobolev Spaces and PDE, page 91), we have for all $\phi\in C^{\infty}_{C}(\mathbb{R}^{n})$ 
\begin{eqnarray}
 \int_{\mathbb{R}^{n}} \int_{\mathbb{R}^{n}}f(x-y)\phi(y)dydx &=& \int_{\mathbb{R}^{n}} \int_{\mathbb{R}^{n}}f(x-y)\phi(y)dxdy      \nonumber \\
   &=& \|\phi\|_{L^{1}(\mathbb{R}^{n})}\|f\|_{L^{1}(\mathbb{R}^{n})}\\
&=& 0
\end{eqnarray}
Therefore, $f=0$.
