On theories of random variables

Abstract

We study theories of spaces of random variables: first, we consider random variables with values in the interval $[0,1]$, then with values in an arbitrary metric structure, generalising Keisler's randomisation of classical structures. We prove preservation and non-preservation results for model theoretic properties under this construction: i) The randomisation of a stable structure is stable. ii) The randomisation of a simple unstable structure is not simple. We also prove that in the randomised structure, every type is a Lascar type.

References in corpus (7)

• Continuous first order logic and local stability
• Topometric spaces and perturbations of metric structures
• Randomizations of models as metric structures
• Modular functionals and perturbations of Nakano spaces
• A proof of completeness for continuous first-order logic
• Continuous and Random Vapnik-Chervonenkis Classes
• Definability of groups in ${\aleph }_0$-stable metric structures

Cited by in corpus (17)

• Randomizations of models as metric structures
• The Real truth
• Stability, NIP, and NSOP; Model Theoretic Properties of Formulas via Topological Properties of Function Spaces
• Continuous Sentences Preserved Under Reduced Products
• Pseudofinite and pseudocompact metric structures
• Propositional Logics for the Lawvere Quantale
• Hoops, Coops and the Algebraic Semantics of Continuous Logic
• Independence relations in randomizations
• Preservation of elementarity by tensor products of tracial von Neumann algebras
• On uniform canonical bases in $L_p$ lattices and other metric structures
• Elementary equivalence and disintegration of tracial von Neumann algebras
• Quantum Expanders and Quantifier Reduction for Tracial von Neumann Algebras
• Vaught's Two-Cardinal Theorem and Quasi-Minimality in Continuous Logic
• Generic Stability and Modes of Convergence
• Stability in affine logic
• Definable closure in randomizations
• Continuous logic and the strict order property