SATURATED STRUCTURES FROM PROBABILITY THEORY Song, Shichang;
In the setting of continuous logic, we study atomless probability spaces and atomless random variable structures. We characterize -saturated atomless probability spaces and -saturated atomless random variable structures for every infinite cardinal . Moreover, -saturated and strongly -homogeneous atomless probability spaces and -saturated and strongly -homogeneous atomless random variable structures are characterized for every infinite cardinal . For atomless probability spaces, we prove that -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.
I. Ben Yaacov, Schrodinger's cat, Israel J. Math. 153 (2006), 157-191.
I. Ben Yaacov, On theories of random variables, Israel J. Math. 194 (2013), no. 2, 957-1012.
I. Ben Yaacov, A. Berenstein, and C. W. Henson, Almost indiscernible sequences and convergence of canonical bases, J. Symb. Log. 79 (2014), no. 2, 460-484.
I. Ben Yaacov, A. Berenstein, C. W. Henson, and A. Usvyatsov, Model theory for metric structures, in: Model Theory with Applications to Algebra and Analysis, Volume 2, 315-427, London Math. Society Lecture Note Series, 350, Cambridge University Press, 2008.
I. Ben Yaacov and A. Usvyatsov, On d-finiteness in continuous structures, Fund. Math. 194 (2007), no. 1, 67-88.
I. Ben Yaacov and A. Usvyatsov, Continuous first order logic and local stability, Trans. Amer. Math. Soc. 362 (2010), no. 10, 5213-5259.
A. Berenstein and C. W. Henson, Model theory of probability spaces, submitted.
I. Berkes and H. P. Rosenthal, Almost exchangeable sequences of random variables, Z. Wahrsch. Verw. Gebiete 70 (1985), no. 4, 473-507.
R. Durrett, Probability: Theory and Examples, Second Edition, Duxbury Press, 1995.
S. Fajardo and H. J. Keisler, Model Theory of Stochastic Processes, Lecture Notes in Logic, 14, Association for Symbolic Logic, 2002.
D. H. Fremlin, Measure algebras, in: Handbook of Boolean Algebras, Volume 3, 877-980, North-Holland, 1989.
D. H. Fremlin, Measure algebras, Volume 3 of Measure Theory, Torres Fremlin, 2003-04; for information see http://www.essex.ac.uk/maths/staff/fremlin/mt.htm.
D. Hoover, A characterization of adapted distribution, Ann. Probab. 15 (1987), no. 4, 1600-1611.
D. Hoover and H. J. Keisler, Adapted probability distributions, Trans. Amer. Math. Soc. 286 (1984), no. 1, 159-201.
H. J. Keisler and Y. Sun, Why saturated probability spaces are necessary, Adv. Math. 221 (2009), no. 5, 1584-1607.
D. Maharam, On homogeneous measure algebras, Proc. Natl. Acad. Sci. USA 28 (1942), 108-111.
S. Song, Model theory and probability, Ph.D. thesis, University of Illinois at Urbana-Champaign, 2011.
S. Song, On d-finite tuples in random variable structures, Fund. Math. 221 (2013), no. 3, 221-230.
S. Song, Axioms for the theory of random variable structures: an elementary approach, J. Korean. Math. Soc. 51 (2014), no. 3, 527-543.