Is the positive cone of $L^p(\Omega)$ weak-closed?

Question

Let $(\Omega,\mu)$ be a $\sigma$-finite measure space. Suppose $1 \leq p< \infty$. Consider the cone $L^p(\Omega)_+$ of positive functions of $L^p(\Omega)$.

Is $L^p(\Omega)_+$ weak-closed in $L^p(\Omega)$ ?

Answer 1

For $p>1$, $L_p$ is reflexive and this set is convex so it is weakly closed because it is norm-closed (Mazur's lemma).

In $L_1$, suppose that $(f_n)_{n=1}^\infty$ is a sequence of non-negative functions converging weakly to $f$. Suppose that $f$ is strictly negative on a set $A$ of positive measure. Let $g$ be the indicator function of $A$. We have

$$\int\limits_\Omega f_n g = \int\limits_A f_n \to \int\limits_\Omega f g= \int\limits_A f < 0.$$

This is impossible as $f_n$ are non-negative. This argument will work in $L_p$ too, so Mazur's lemma is not quite necessary.