Grabisch: A general discrete Choquet integral

We consider a collection F of subsets of a finite set N together with a capacity v: F → R+ and call a function f: N → R measurable if its level sets belong to F. Only in this case, the classical definition of the Choquet integral works in our wider context. In this article, we provide a general framework for a Choquet integral that works also for non-measurable functions f and includes the integral proposed by Lehrer as a special case. We show that the general Choquet integral can be computed by a Monge-type algorithm. Moreover, we derive several properties that relate to the extension of capacities on 2N and to supermodularity, in particular