The author has generalized the classical measure theory by considering, as the domain of a measure function, not an algebra, but a \(\sigma\)-base, i.e. a collection of sets closed under complementation and countable unions of pairwise disjoint sets only. The theory may be more adequate than the usual one for modelling probability in quantum physics. The present paper is a survey of the theory with many results having already been published elsewhere. The generalized theory is motivated and developed. As the extended theory is in several ways different from the classical one questions like embedding a generalized measure space into a classical measure space and additivity of the integral are discussed. Also, quantum logics and Hilbert space quantum mechanics are considered and compared to the extended measure theory. Finally, an application to a deterministic spin model is presented.