Isometry group of Borel randomizations

TL;DR

This paper investigates the isometry group of Borel randomizations of structures, demonstrating that key dynamical properties like extreme amenability are preserved from the original structure to its randomization.

Contribution

It establishes that properties such as the Rohklin property and extreme amenability transfer from a structure's isometry group to its Borel randomization, revealing new invariance results.

Findings

Rohklin property holds for the isometry group of the randomization

Extreme amenability is preserved in the randomization context

Topometric generics transfer to the randomized setting

Abstract

We study global dynamical properties of the isometry group of the Borel randomization of a separable complete structure. In particular, we show that if properties such as the Rohklin property, topometric generics, extreme amenability hold for the isometry group of the structure, they also hold in the isometry group of the randomization.