Аннотация:
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 Xμ∼Y if μ(X△Y)=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 ◻ as the aforementioned interior operator.
In this paper we prove a general completeness result which implies the following two facts:
1. the logic S4u is complete for interpretations on any subset of Euclidean space of positive Lebesgue measure;
2. 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.