Is the set of positive function set in $H^{-1}(\Omega)$ closed?

Question

Let $\Omega\subset\mathbb R^n$ be a bounded and open domain with smooth boundary. Define the set $$ A:=\{u\in H^{-1}(\Omega)~|~u(x)\geq0,~a.e.~x\in\Omega\}. $$ Is the set $A$ closed?

Answer 1

Yes. If you identify $H^{-1}$ to $H^1_0$ (you can always do that by Riesz representation theorem), you can consider $f \in \text{cl }A$ and $(f_n)_n \subset A$ such that $f_n \to f$ in $H^1_0$. In particular $f_n \to f$ in $L^2$ and the exists a subsequence (still denoted $f_n$) such that $f_n \to f$ almost everywhere. This implies that $f \ge 0$ a.e.