Integrals based on monotone set functions

An overview of various integrals is given which can be defined on arbitrary monotone set functions vanishing in the empty set (called here monotone measures). Our survey includes not only the Choquet integral (1954) [10], the Shilkret integral (1971) [66] and the Sugeno integral (1974) [71] and some of their properties, but also some more general and more recent concepts as universal integrals Klement et al. (2010) [27] and decomposition integrals Even (2014) [13], together with some of their properties, such as integral inequalities and convergence theorems.