Following the failure of the previous attempt at axiomatizing the envisaged extension of FOL with almost-everywhere quantification endowed with measure-theoretic semantics, a two-sorted solution is presented. The additional sort is used to deal with measurable sets of individuals. The new axioms for dealing with the almost-everywhere quantification are discussed. The completeness of the new axiomatization is carried out using the Lindenbaum-Henkin technique. Joint work with Luís Cruz-Filipe, Cristina Sernadas and Amílcar Sernadas.