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$f,g \in L^1 (\Omega)$, $\Omega\subset\mathbb{R}^n$ 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

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    @TTT, i wass thinking on this question and i think you have to change $\Omega$ by $\mathbb{R}^{n}$, because $f(x-y)$ is not well defined for all $x\in\Omega$.2012-10-20

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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, $f=0$.