ABSTRACT
The Homogeneity Theorem from Chapter 2 showed us that in a hyperfinite adapted space two processes with the same adapted distribution are the same in a precise mathematical sense. They are automorphic, that is, there is an adapted isomorphism from the space to itself which carries one process to the other almost surely. The notion of automorphism used is very strong-one process is essentially a renaming of the other. What can be said of processes that live on possibly different adapted spaces and have the same adapted distribution? Do we have any result that resembles the strong characterization of adapted equivalence in a hyperfinite adapted space? There is an answer: they are isomorphic, but in a weaker sense. In Section 5A we will prove the Intrinsic Isomorphism Theorem, which makes this statement rigorous. It is due to Hoover and Keisler, but was first published in Section 3 of Fajardo [1987] with some minor generalizations. This result was used in Hoover [1992]. Section 5B is based on both of the papers Fajardo [1987] and Hoover [1992], and uses the notions introduced in Section 5A for a theory of definability in adapted spaces.
