Generic separable metric structures

TL;DR

This paper compares different notions of genericity in separable metric structures, offering a model theoretic approach to understand their topological and measure-theoretic properties, with applications to Urysohn's space.

Contribution

It introduces a general model theoretic technique to analyze genericity in both topological and measure-theoretic contexts for separable metric structures.

Findings

Provides a new perspective on Vershik's theorems

Develops a unified approach to genericity

Connects model theory with descriptive set theory

Abstract

We compare three notions of genericity of separable metric structures. Our analysis provides a general model theoretic technique of showing that structures are generic in descriptive set theoretic (topological) sense and in measure theoretic sense. In particular, it gives a new perspective on Vershik's theorems on genericity and randomness of Urysohn's space among separable metric spaces.