How do I show, that
$\text{Let $\Omega \subset \mathbb{R}^n$ be open. Satisfies } f\in\mathcal{L}^1(\Omega) \text{ following propertie }$ $$\int_{\Omega} f(x)g(x) dx = 0 \text{ for all } g\in\mathcal{C}_c^0(\Omega),$$ $\text{then } f = 0 \text{ almost everywhere in } \Omega. \text{ Whereby } \mathcal{C}_c^0(\Omega):= \{f \in \mathcal{C}^0(\Omega): supp\text{ }f \text{ is compact}\}.$
A hint was given, that functions like $g = \frac{h}{\sqrt{1+h^2}}$ for suitable $h \in \mathcal{C}_c^0(\Omega)$ were helpfull.