Completeness of \(\mathrm S4\) for the Lebesgue measure algebra (Q427238)

The author shows that the modal logic S4 is complete with respect to the Lebesgue measure algebra \(\mathcal M\), which is the quotient of the Lebesgue-measurable subsets of the unit interval by the ideal of subsets of Lebesgue measure zero. This is done by translating refutations on the infinite binary tree \(T_2\) to refutations on \(\mathcal M\), and using the well-known completeness of S4 with respect to \(T_2\). This result has also been independently obtained by \textit{D. Fernández-Duque} [in: Advances in modal logic. Vol. 8. Proceedings of the 8th conference (AiML 2010), Moscow, Russia, August 24--27, 2010. London: College Publications. 100--119 (2010; Zbl 1254.03031)].