On theories of random variables

TL;DR

This paper investigates the properties of spaces of random variables within model theory, focusing on how certain properties are preserved or not under the randomisation process, including stability and simplicity.

Contribution

It generalizes Keisler's randomisation to arbitrary metric structures and establishes key preservation and non-preservation results for model-theoretic properties.

Findings

Randomisation of a stable structure remains stable.

Randomisation of a simple unstable structure is not simple.

In the randomised structure, all types are Lascar types.

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.