Independence relations in randomizations

TL;DR

This paper investigates different notions of independence within models of the randomized theory of a complete first order theory, exploring how these concepts behave in a probabilistic setting.

Contribution

It introduces and analyzes various independence relations in the context of the randomization of a complete first order theory, expanding understanding of probabilistic model theory.

Findings

Different independence notions are characterized in models of the randomized theory.

The properties and relationships of these independence notions are systematically studied.

Results contribute to the understanding of probabilistic independence in model theory.

Abstract

The randomization of a complete first order theory $T$ is the complete continuous theory $T^R$ with two sorts, a sort for random elements of models of $T$, and a sort for events in an underlying probability space. We study various notions of independence in models of $T^R$.