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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.

monoid
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  • It is worth noting with the hint that when $|h|$ is close to zero, $g\approx h$. However, when $|h|>>0$, $g\approx 1$. So the hint is essentially to consider bump functions. – Aaron Nov 29 '15 at 09:19
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    Let $A={x:f(x)>0}$. Show that if $\mu(A)>0$, you can find $g$ s.t. $\int_A fg>0$. – A.S. Nov 29 '15 at 09:20

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