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 2 and to supermodularity, in particular.