Quantified S4 in the Lebesgue measure algebra with a constant countable domain

Define quantified S4, QS4 [first-order S4, FOS4], by combining the axioms and rules of inference of propositional S4 with the axioms and rules of classical first order logic without identity [with identity]. In the 1950’s, Rasiowa and Sikorski extended the algebraic semantics for propositional S4 to a constant-domain algebraic semantics for QS4, and showed that QS4 is sound and complete for this semantics. Recently, Lando has extended the algebraic semantics for propositional S4 to an expanding-domain algebraic semantics FOS4. Her main result is that FOS4 is complete for an algebra of particular interest, the Lebesgue measure algebra, with expanding countable domains. In the current paper, we show that QS4, without identity, is complete for the Lebesgue measure algebra with a constant countable domain. One takeaway is that measure-theoretic semantics might need varying domains to handle identity but does not need them to handle quantification.