Saturated Structures From Probability Theory

Abstract

Abstract. In the setting of continuous logic, we study atomless probability spaces and atomless random variable structures. We characterize κ-saturated atomless probability spaces and κ-saturated atomless random variable structures for every infinite cardinal κ. Moreover, κ-saturated and strongly κ-homogeneous atomless probability spaces and κ-saturated and strongly κ-homogeneous atomless random variable structures are characterized for every infinite cardinal κ. For atomless probability spaces, we prove that ℵ 1 -saturation is equivalent to Hoover-Keisler saturation. For atomless random variable structures whose underlying probability spaces are Hoover-Keisler saturated, we prove several equivalent conditions.

Model theory of probability spaces

“…Note that for each infinite cardinal κ, we identified the κ-saturated model of AP A of density character κ as the Maharam homogenous model of density κ. More information on κ-saturated and κ-homogeneous models of AP A can be found in [26].…”