Purification, Saturation and the Exact Law of Large Numbers

Purification results are important in game theory and statistical decision theory. The purpose of this paper is to prove a general purification theorem that generalizes many authors' results. The key idea of our proof is to make use of the exact law of large numbers. As an application, we show that every mixed strategy in games with finite players, general action spaces, and diffused, conditionally independent incomplete information has many strong purifications.