Denote with $C_0(\Omega)$ the subspace of countinuous functions with compact support, $\Omega\subseteq\mathbb{R}^n$ open. Define an application on $C_0(\Omega)$ as $||\cdot||\colon C_0(\Omega)\to\mathbb{R}_+$ $$||f||_1:=\int_\Omega |f|\;d\lambda_n$$ where $\lambda_n$ is the Lebesgue measure.
I must be prove that $||\cdot||_1$ is a norm on $C_0(\Omega)$. The triangle inequality and positivity is ok. Remain to prove that $$||f||_1=0\iff f=0\;\text{on}\;\Omega.$$
$(\Leftarrow)$ is obvious; I am having difficulty proving $(\Rightarrow)$.
Suppose that $$||f||_1=0\Rightarrow f=0\;\text{a.e in support}$$
Can I conclude?
If $f(x_0)\neq 0$ for some $x_0\in\Omega$ then since $f$ is continuous there must be a neighborhood $U$ of $x_0$ where $f(x)\neq 0$ for all $x\in U$, and now for an $\varepsilon>0$ there is a ball $B_{\varepsilon}(x_0)\subset U$ .Clearly $$0<\int_{B_{\varepsilon}(x_0 )}|f|d\lambda_n\leq\int_{\Omega}|f|d\lambda_n$$