Saturated structures from probability theory
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In the setting of continuous logic, we study atomless probability spaces and atomless random variable structures. We characterize $\kappa$-saturated atomless probability spaces and $\kappa$-saturated atomless random variable structures for every infinite cardinal $\kappa$. Moreover, $\kappa$-saturated and strongly $\kappa$-homogeneous atomless probability spaces and $\kappa$-saturated and strongly $\kappa$-homogeneous atomless random variable structures are characterized for every infinite cardinal $\kappa$. \!For atomless probability spaces, we prove that $\aleph_1$-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.
Abstract
In the setting of continuous logic, we study atomless probability spaces and atomless random variable structures. We characterize $\kappa$-saturated atomless probability spaces and $\kappa$-saturated atomless random variable structures for every infinite cardinal $\kappa$. Moreover, $\kappa$-saturated and strongly $\kappa$-homogeneous atomless probability spaces and $\kappa$-saturated and strongly $\kappa$-homogeneous atomless random variable structures are characterized for every infinite cardinal $\kappa$. \!For atomless probability spaces, we prove that $\aleph_1$-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.
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