Absolute Completeness of S4u for Its Measure-Theoretic Semantics

Given a measure space 〈X,μ〉, we define its measure algebra Aμ as the quotient of the algebra of all measurable subsets of X modulo the relation X μ ∼ Y if μ(X4Y ) = 0. If further X is endowed with a topology T , we can define an interior operator on Aμ analogous to the interior operator on P(X). Formulas of S4u (the modal logic S4 with a universal modality ∀ added) can then be assigned elements of Aμ by interpreting 2 as the aforementioned interior operator. In this paper we prove a general completeness result which implies the following two facts: (i) the logic S4u is complete for interpretations on any subset of Euclidean space of positive Lebesgue measure; (ii) the logic S4u is complete for interpretations on the Cantor set equipped with its appropriate fractal measure. Further, our result implies in both cases that given ε > 0, a satisfiable formula can be satisfied everywhere except in a region of measure at most ε.