Suppose $(X,\Delta\ge 0)$ is a pair such that $(p^g-1)(K_X+\Delta)$ is an Integral Weil Divisor for some $g>0$ and $X$ is a normal variety. Define $\mathcal{L}_{e,\Delta} = \mathcal{O}_X( (1-p^g)(K_X + \Delta) )$ and note it is a reflexive sheaf.

We can still define the map $\phi^e: \mathcal{L}_{e, \Delta} \to \mathcal{O}_X$ for $g|e$, following the Grothendieck Trace map, Reflexivity of Weil divisors and normality of $X$. Now my question is, can I define the *non-$F$-pure* ideal $\sigma(X, \Delta)$ of $(X, \Delta$) as $\sigma(X, \Delta)=\bigcap_{e\ge 0} \phi^{eg}F^{eg}_*\mathcal{L}_{e, \Delta}$ ?

What I mean is that, will this definition have all the good properties of the $\mathbb{Q}$-Cartier case? For example, I need to know whether $\phi^{eg}F^{eg}_*\mathcal{L}_{e, \Delta}=\sigma(X, \Delta)$ for all $e\gg 0$ or not?

Also, is $\sigma(X, \Delta)$ the Unique Largest ideal $\mathcal{I}$ such that $\phi^{eg}F^{eg}_*(\mathcal{L}_{e, \Delta}\cdot \mathcal{I})=\mathcal{I}$ for all $e>0$?