On Quadruplets of Nonadditive Integrals

The definition of the well-known Choquet integral of a nonnegative random variable is based on a survival (decumulative distribution) function. A quadruplet of Choquet integrals is obtained by extending this idea also to a (cumulative) distribution function for real random variables. The concept includes the asymmetric and symmetric Choquet integrals as well as their complementary versions. These four integrals coincide for any additive measure. For a finite measure, the complementary integrals are related to the original asymmetric and symmetric Choquet integrals via dual measure. Duality also plays a key role in the study of basic integral properties. An alternative way to look at Choquet integral goes via the area of a survival function subgraph. Substituting the Lebesgue planar measure with an arbitrary planar (fuzzy) measure we generalize the introduced quadruplet concept in which the Sugeno and the Shilkret integral quadruplets are involved. This gives a new insight into a common generalization of the three most known nonadditive integrals of Choquet, Sugeno and Shilkret as well as their symmetric, asymmetric and complementary versions.