Sufficient Statistics and Extreme Points

A convex set M is called a simplex if there exists a subset Mₑ of M such that every P ∈ M is the barycentre of one and only one probability measure μ concentrated on Mₑ. Elements of Mₑ are called extreme points of M. To prove that a set of functions or measures is a simplex, usually the Choquet theorem on extreme points of convex sets in linear topological spaces is cited. We prove a simpler theorem which is more convenient for many applications. Instead of topological considerations, this theorem makes use of the concept of sufficient statistics.