This Handbook chapter brings an exhaustive overview of pseudo-additive mesures, corresponding integrals, but first of all of several possible applications. The chapter consists of six sections. In the first one, pseudo-operations are introduced and the basics of the pseudo-additive measure theory (including integral, convolution, Riesz type theorem, etc.) are recalled. The second section discusses some applications of the pseudo-additive measure theory in different fields, such as the probabilistic metric spaces, information theory, system theory, etc. For example, the addition of fuzzy numbers is shown to be a special type of pseudo-convolution of possibility distributions. Pseudo-operations based on a generator \(g\) are treated in the third section, including the corresponding measures and integrals. Special limits of \(g\)-integrals are shown to be idempotent integrals, indicating the links between the Archimedean and the idempotent analysis. Very interesting applications of the pseudo-additive measure theory in the field of nonlinear partial differential equations (PDE) are discussed in the fourth section. Recall here some special cases: the Hamilton-Jacobi equation with non-smooth Hamiltonian (including Burgers PDE) and the Bellman equation. A possibility generalization of pseudo-operations violating the commutativity (by pseudo-addition and pseudo-multiplication) is introduced in Section 5, indicating possible applications in PDE area and discussing the corresponding pseudo-measure. Finally, the last section deals with conditional distributive real semirings. Up to the isomorphism, these are shown to be mixtures (ordinal sums) of idempotent semirings and the standard semiring of nonnegative reals. The corresponding hybrid probability-possibilistic measures and integrals are exploited to build up a hybrid utility theory. The chapter brings also a huge references list for readers interested in more details. It can be recommended to any researcher interested in non-probabilistic modeling and non-classical analysis.
For the entire collection see [Zbl 0998.28001].