Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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J. Korean Math. Soc. 2016; 53(2): 315-329

Printed March 1, 2016

https://doi.org/10.4134/JKMS.2016.53.2.315

Copyright © The Korean Mathematical Society.

Saturated structures from probability theory

Shichang Song

Beijing Jiaotong University

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.

Keywords: continuous logic, saturation, Maharam spectrum, probability algebras, random variables

MSC numbers: 03C50, 28A60