On the extendibility of finitely exchangeable probability measures

We give necessary and sufficient conditions in order that a finite sequence $(X_1, \ldots, X_n)$ of exchangeable random elements in a fairly general space $S$ be extendible to a longer finite or to an infinite exchangeable sequence. This is done by formulating the extendibility problem as the extension problem for certain bounded linear functionals on suitable normed spaces and by using the Hahn-Banach theorem and other functional and measure-theoretic techniques. We examine when such a finitely exchangeable random sequence is a mixture (with respect to a probability measure) of product measures and also study the preservation of the extendibility property under suitable limiting operations.