Finitely additive beliefs and universal type spaces

In this paper we examine the existence of a universal (to be precise: terminal) type space when beliefs are described by finitely additive probability measures. We find that in the category of all type spaces that satisfy certain measurability conditions (-measurability, for some fixed regular cardinal ), there is a universal type space (i.e. a terminal object, that is a type space to which every type space can be mapped in a unique beliefs-preserving way (the morphisms of our category, the so-called type morphisms)), while, by an probabilistic adaption of the elegant sober-drunk example of Heifetz and Samet (1998a), we show that if all subsets of the spaces are required to be measurable there is no universal type space.