A solution to a conjecture of David Schmeidler

The purpose of this paper is to disprove an old and interesting conjecture in the theory of cooperative games initially presented as a reasonable conjecture in the seminal work of Schmeidler (J Math Anal Appl 40:214–225, 1972): An exact capacity continuous at the empty set has a countably additive probability in its core. In this paper, we show that this conjecture is not true in general. More precisely, we give a large class of topological spaces for which the conjecture fails, even with the stronger assumption that the capacity is convex. However, on ( N , 2 N ) the conjecture holds for convex capacities, as it is easy to show. We prove that in its original form, that is for exact capacities, it fails.