ABSTRACT
In the preceding chapters we have often used the adjective “rich” to describe the powerful properties of saturated and hyperfinite adapted spaces. In this chapter we formally introduce the mathematical notion of a rich adapted space and prove three main theorems about them (in these theorems we allow filtrations which are not necessarily right continuous): (1) With a countable time line, an adapted space is rich if and only if it is saturated. (2) Every atomless adapted Loeb space is rich. (3) For every rich adapted space with the real time line, the corresponding right continuous adapted space is saturated. This will show that all atomless adapted Loeb spaces are saturated. In the next chapter we will see that rich adapted spaces have some strong properties which quickly lead to a variety of applications in probability theory.
