The extreme points of the set of belief measures

It is known that the σ-additive Möbius transform of a belief function (we prefer to call it belief measure) can be derived from Choquet’s Theorem. One has to show that the extreme points of the compact convex set of belief measures are the {0, 1}-valued belief measures, which are called filter games as well. A proof is implicit in the famous 1953/54 paper of Choquet but it is hard to read it. We present a direct proof and – for the sake of completeness – derive the Möbius transform.