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Assume that $f\in {L}_{\text{loc}}^{1}\left({\mathbb{R}}^{n},\mathbb{R}\right)$ is such that
${\int }_{{\mathbb{R}}^{n}}f\left(x\right)g\left(x\right)dx\ge 0$
for all $g\in {C}_{c}\left({\mathbb{R}}^{n},{\mathbb{R}}^{+}\right).$ Here ${C}_{c}\left({\mathbb{R}}^{n},{\mathbb{R}}^{+}\right)$ denotes the set of all continuous functions on ${\mathbb{R}}^{n}$ taking non-negative real values and which have compact support. I think it follows that $f\left(x\right)\ge 0$ for a.e. $x\in {\mathbb{R}}^{n}$. However I only managed to prove this implication in the case where $f$ is (essentially) bounded. How to go from this simpler case to the general case ?
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Kiana Cantu
Fix a family of mollifier $\left({g}_{ϵ}{\right)}_{ϵ}>0$. For any $R>0$ let

where ${B}_{R+1}$ is the ball of radius $R+1$ in ${\mathbb{R}}^{n}$. Then ${f}_{R}\ast {g}_{ϵ}$ converges in ${L}^{1}\left({\mathbb{R}}^{n}\right)$ to ${f}_{R}$. Restricting to ${B}_{R}$, ${f}_{R}\ast {g}_{ϵ}$ converges in ${L}^{1}\left({B}_{R}\right)$ to ${f}_{R}=f$.
Also, if $ϵ<1$, then the condition implies that

Together with the convergence of ${f}_{R}\ast {g}_{ϵ}\to f$ in ${L}^{1}\left({B}_{R}\right)$, there is a sequence ${ϵ}_{n}\to 0$ so that
${f}_{R}\ast {g}_{{ϵ}_{n}}\to f$
a.e. in ${B}_{R}$. Thus $f\ge 0$ in ${B}_{R}$. Since $R>0$ is arbitrary, choosing $R=K$. Using
$\left\{x\in {\mathbb{R}}^{n}:f\left(x\right)<0\right\}=\bigcup _{K\in \mathbb{N}}\left\{x\in {B}_{K}:f\left(x\right)<0\right\},$
one also has $f\ge 0$ a.e. in ${\mathbb{R}}^{n}$.