Test Functions and positiveness of a functions

Question

Let $f\in L_{\text{loc}}^1(\Omega)$ such that for every nonnegative test function $\phi\in C^0_c(\Omega)$ $$ \int_\Omega f\phi\geq 0.$$ How can one proof that $f\geq 0\;\;a.e$?

Answer 1

Hint: WLOG $f\in L^1(\Omega).$ Find $g_n \in C_c(\Omega)$ such that $|g_n|\le 1$ on $\Omega$ and $g_n \to \text { sgn }(f)$ a.e.