Let $L^\infty_+$ be the set of all $f\in L^\infty$ which are non negative. Our measure can be assumed to be finite. My goal is to prove that $L^\infty_+$ is closed in the weak-star topology $\sigma(L^\infty,L^1)$. Hence I view $L^\infty$ as the dual of $L^1$. We know that every linear continuous function of $L^1$ can be written:
$$F(x)=\int xy d\mu$$
for $x\in L^1$ and a unique $y\in L^\infty$. So $L^\infty_+$ is the space of all $F$ with $y\ge 0\iff \int xy d\mu \ge 0 $ for all $0\le x\in L^1$. I've learned that the weak-star topology (on $L^\infty$) is generated by the sets
$$\Omega_{x,U}=\{y\in L^\infty: \int xy d\mu\subset U \}$$
for $U\subset \mathbb{R}$ closed. Somehow I have to choose $x,U$ such that $\Omega_{x,U}=L^\infty_+$. But maybe there is a simpler characterization of closed sets in $\sigma(L^\infty,L^1)$? Or how else could I show that $L^\infty_+$ is weak-star closed?
One can write $$L^\infty_+=\bigcap_{A\in\mathcal F}\left\{g\in L^{\infty},\int_Ag\geqslant 0\right\},$$ where $\mathcal F$ is the collection of sets of finite measure. Inclusion $\subset$ is obvious, and $\supset$ follows by the following argument. Assume the measure space $\sigma$-finite (hence $X=\bigcup F_n$, where each $F_n$ is measurable and has finite measure, and the $F_n$ are nested); if $g<0$ on a set of positive measure, take the intersection of the characteristic function of this set with a $F_n$ for a $n$ large enough.
And for each $A\in\mathcal F$, $\left\{g\in L^{\infty},\int_Ag\geqslant 0\right\}$ is weak-* closed, and so is an intersection of such sets.