The LindstrØm-Type Characterization of Hajek's Fuzzy Logic of Integrals

In 1969, Per LindstrØm proved his famous theorem and established criteria for the first-order definability of formal theories for discrete structures. The results were extrapolated for systems of modal logic and even for theories for continuous structures. This paper aims to formulate and prove LindstrØm's theorem for analytic structures based on measures. In particular, Hajek's Logic of Integral is redefined as an abstract logic with a new type of Hajek's satisfiability and considered as a minimal logic in the class of analytic structures with Lebesgue integrals.