Monotone measures–based integrals: special functionals and an optimization tool

Considering the space of all non–negative measurable functions F(X,A) linked to a fixed measurable space (X,A), the Lebesgue integral can be seen as an additive, continuous from below functional L on F(X,A), related to a measure m : A → [0,∞] given by m(A) = L(1A). We introduce several other integrals which can be seen as special functionals on F(X,A). For example, the Choquet integral [7] is a comonotone additive functional C, and the corresponding monotone measure m : A → [0,∞] is given by m(A) = C(1A). Note that though the first traces of Choquet integral goes back to Vitali [18], its comonotone additivity was stressed by [15, 16] based on inputs from economy. Similarly, the Sugeno integral Su [17] is a min–homogeneous comonotone maxitive functional on F(X,A). Among several approaches to integration with respect to monotone measures, covering both the Choquet and the Sugeno integrals, we recall the axiomatic approach proposed by Benvenuti et al. [1]. A rather general framework of universal integrals was recently proposed in [8], unifying the look on both Choquet and Sugeno integral. Two alternative looks on universal integrals are offered in our paper [5].