Every hierarchy of beliefs is a type

When modeling game situations of incomplete information one usually considers the players' hierarchies of beliefs, a source of all sorts of complications. Hars\'anyi (1967-68)'s idea henceforth referred to as the "Hars\'anyi program" is that hierarchies of beliefs can be replaced by "types". The types constitute the "type space". In the purely measurable framework Heifetz and Samet (1998) formalize the concept of type spaces and prove the existence and the uniqueness of a universal type space. Meier (2001) shows that the purely measurable universal type space is complete, i.e., it is a consistent object. With the aim of adding the finishing touch to these results, we will prove in this paper that in the purely measurable framework every hierarchy of beliefs can be represented by a unique element of the complete universal type space.