Absolute completeness of $\mathsf{S4}_u$ for its measure-theoretic semantics

Given a measure space $\langle X,\mu\rangle$ we define its measure algebra $\mathbb A_\mu$ as the quotient of the algebra of all measurable subsets of $X$ modulo the relation $X\overset\mu\sim Y$ if $\mu(X\triangle Y)=0$. If further $X$ is endowed with a topology $\mathcal T$ , we can define an interior operator on $\mathbb A_\mu$ analogous to the interior operator on $\mathcal P(X)$. Formulas of $\mathsf{S4}_u$ (the modal logic $\mathsf{S4}$ with a universal modality $\forall$ added) can then be assigned elements of $\mathbb A_\mu$ by interpreting $\square$ as the aforementioned interior operator.
In this paper we prove a general completeness result which implies the following two facts:

• 1. the logic $\mathsf{S4}_u$ is complete for interpretations on any subset of Euclidean space of positive Lebesgue measure;
• 2. the logic $\mathsf{S4}_u$ is complete for interpretations on the Cantor set equipped with its appropriate fractal measure.

Further, our result implies in both cases that given $\varepsilon>0$, a satisfiable formula can be satisfied everywhere except in a region of measure at most $\varepsilon$.