We give an axiomatization of first-order logic enriched with the almost-everywhere quantifier over finitely additive measures. Using an adapted version of the consistency property adequate for dealing with this generalized quantifier, we show that such a logic is both strongly complete and enjoys Craig interpolation, relying on a (countable) model existence theorem. We also discuss possible extensions of these results to the almost-everywhere quantifier over countably additive measures. The talk reports on joint work with Wafik Lotfallah and Cristina Sernadas.