When does $0=\int f(x)\psi(x)\,dx$ for all $f\in L^1(G)$ imply that $\psi=0$

Question

Let $G$ be a locally compact group with Haar measure $dx$.

I am wondering what conditions a function $\psi$ must satisfy so that the following holds.

$\int f(x)\psi(x)\,dx=0$ for all $f\in L^1(G)\ \ \ \ \ \ \ \ \Rightarrow\ \ \ \ \ \ \ \ \ $ $\psi=0\ \ (a.e.)$

And how do I proove that the implication is true? I hope someone can help, I have no idea of where to start..