Following recent developments in the topic of generalized quantifiers, and also having in mind subsequent applications to the area of cleistic logic, we propose a conservative enrichment of first-order logic with almost-everywhere quantification, endowed with measure-theoretic semantics. The completeness of the axiomatization we provide is carried out using a variant of the Lindenbaum-Henkin technique. The independence of the axioms is analysed, and the almost-everywhere quantifier is classified using the taxonomy proposed by Carnielli et al. Joint work with J. Rasga and A. Sernadas.