Integrals

Abstract

It is well known that in the case of classical (additive) measures, the Lebesgue integral is the usual definition of an integral with respect to a measure, and it allows the computation of the expected value of random variables. The question which is addressed in this chapter is: How to define the integral of a function with respect to a nonadditive measure, i.e., a capacity or a game? As we will see, the answer is not unique, and there exist many definitions in the literature. Nevertheless, two concepts of integrals emerge: the one proposed by Choquet in 1953, and the one proposed by Sugeno in 1974. Both are based on the decumulative distribution function of the integrand w.r.t. the capacity, the Choquet integral being the area below the decumulative function, and the Sugeno integral being the value at the intersection with the diagonal. Most of the other concepts of integral are also based on the decumulative function, like the Shilkret integral, but other approaches are possible. For example, the concave integral proposed by Lehrer is defined as the lower envelope of a class of concave and positively homogeneous functionals.