Choquet Integration on Set Systems

We present a framework for a general Choquet integral on systems F of measurable sets relative to a finite universeNthat do not necessarily include all nonempty subsets. In this context, many functions become nonmeasurable, and the classical Choquet integral does not apply. By considering a lower approximation by step functions, we arrive at a natural notion of an integral which generalizes the classical Choquet integral and is meaningful for any function. We observe that the Choquet integral of a measurable function can be computed by a Monge-type algorithm and we characterize so-called weakly union-closed systems as those set systems that allow the Monge algorithm to compute the general Choquet integral. In addition, we characterize the superadditivity of the Choquet integral on these systems.