Fremlin's book "Measure theory" 475B defines two monotone operators on the whole power set of $R^n$ denoted essential interior and essential closure:
\begin{eqnarray} X \mapsto int^* X:=\lbrace x\in R^n \quad | \quad \liminf_{\delta \downarrow 0} \frac{\mu_*(B(x,\delta) \cap X)}{\mu(B(x,\delta))} =1 \rbrace \\ X \mapsto cl^* X :=\lbrace x\in R^n \quad | \quad \limsup_{\delta \downarrow 0} \frac{\mu^*(B(x,\delta) \cap X)}{\mu(B(x,\delta))} >0 \rbrace \end{eqnarray} where $\mu$ is Lebesgue measure and $\mu^*$, $\mu_*$ are the induced outer and inner measures. He then shows that $int^*$ preserves intersections and $cl^*$ preserves unions.
I would like to know under what conditions they can be showed to be idempotent. For sure they are idempotent for measurable sets for which upper and lower densities agree, but I would like to know if it is possible to prove idempotence for arbitrary sets.