Let $1\leq p< \infty$ and $E= L^p(\Omega,\mu)$. If $A \subseteq \Omega$ is a measurable set and $T: E \to E$ is defined by $Tf=\mathbf{1}_Af$, then find the quasi-interior points of $\operatorname{Im} T$.
Attempt: Of course as $T$ is a positive operator, so $\operatorname{Im} T$ is a Banach lattice. I was able to show that $f$ is a quasi-interior point of $E$ if and only if $f(x) >0$ for almost all $x$.
However, I don't think that if $f \in \operatorname{Im} T$ is a quasi-interior point of $\operatorname{Im} T$, then it must be a quasi-interior point of $E$ too. Are quasi-interior point of $\operatorname{Im} T$ just all $f\in \operatorname{Im} T$ such that $f(x)=0$ for almost all $x \in A$?