We investigate a modal extension of the infinitary classical logic with countable meets and joins, formulated with an eye toward measure-theoretic work in dynamical systems and in point-free ergodic theory.We define a modal formalism in this language, which we call modal measurable logics.We also introduce a Kripke-like semantics for these logics in measurable spaces taking a designated modal -ideal into consideration.Using a restriction of Jónsson-Tarski duality and a modal extension of the Loomis-Sikorski theorem, we prove completeness of modal measurable logics with respect to this new semantics.
Bezhanishvili et al. (Sun,) studied this question.