Positivity of the weak * limit in $L^{\infty}$. [closed]

Assume $\{f_n\}$ is the sequence where $0 \leq f_n \in L^{\infty} (\mathbb{R}^3)$. Now suppose $f$ is the weak * limit of $\{f_n\}$ in $L^{\infty} (\mathbb{R}^3)$.

Can I derive $f \geq 0$?

I think it might be true by taking the sign function on measurable sets.

You can deduce that
$$\tag1 \int_{\mathbb R^3} fg\geq0\qquad \text{ for all } g\in L^1(\mathbb R^3),\ g\geq0. $$ This is enough to show that $f\geq0$ a.e.: if $f<0$ on any finite-measure set $E$, then if $E$ has positive measure you get from $(1)$
$$ 0>\int_Ef=\int f\,1_E\geq0, $$ a contradiction. So $E$ is a nullset, and so $f\geq0 $ a.e.