The Symmetric Choquet Integral with Respect to Riesz-space-valued Capacities

In [3] we introduced a “monotone-type” (that is, Choquet-type) integral for realvalued functions, with respect to finitely additive positive set functions, with values in a Dedekind complete Riesz space. A “Lebesgue-type” integral for such kind of functions was investigated in [7]. In [4] we gave some comparison results for these types of integrals. In [10], a Choquet-type integral for real-valued functions with respect to Rieszspace-valued “capacities”, that is, monotone set functions not necessarily finitely additive, is investigated. The study of these integrals is motivated by several branches of mathematics (for example, stochastic processes, see [16]) and has also some applications to probability theory and economics, for example for the study of the fundamental properties of the so-called “utility functions” (see for instance [14], [19], [21], [22]) and the study of “qualitative probabilities”, that is, set functions which associate to an event not necessarily a real number (indeed, in reality it is often