Families of Measures and their Envelopes

Abstract

Families of measures play important roles in statistics and economic decision theory in order to model uncertainty. They also appear as the core of a cooperative game in game theory. The envelopes (supremum or infimum) of the members of the family are set functions and their integrals give bounds for the integrals in the family. The main result is a characterization of submodular set functions by means of envelopes of additive set functions. The method of generating a set function as supremum of a given family of set functions will be employed, too, for proving the Radon-Nikodym Theorem in the next chapter.