Modal Measurable Logics via a Modal Loomis-Sikorski Representation Theorem
This work provides a foundational logical framework for measure-theoretic work in dynamical systems and point-free ergodic theory.
The paper introduces modal measurable logics, an extension of infinitary classical logic with countable meets and joins, and proves completeness with respect to a Kripke-like semantics in measurable spaces using a modal Loomis-Sikorski representation theorem.
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 sigma-ideal into consideration. Using a restriction of Jonsson-Tarski duality and a modal extension of the Loomis-Sikorski theorem, we prove completeness of modal measurable logics with respect to this new semantics.