INVARIANT MEASURES CONCENTRATED ON COUNTABLE STRUCTURES

Nathanael Ackerman,Cameron Freer,Rehana Patel

doi:10.1017/fms.2016.15

Abstract

Let$L$be a countable language. We say that a countable infinite$L$-structure${\mathcal{M}}$admits an invariant measure when there is a probability measure on the space of$L$-structures with the same underlying set as${\mathcal{M}}$that is invariant under permutations of that set, and that assigns measure one to the isomorphism class of${\mathcal{M}}$. We show that${\mathcal{M}}$admits an invariant measure if and only if it has trivial definable closure, that is, the pointwise stabilizer in$\text{Aut}({\mathcal{M}})$of an arbitrary finite tuple of${\mathcal{M}}$fixes no additional points. When${\mathcal{M}}$is a Fraïssé limit in a relational language, this amounts to requiring that the age of${\mathcal{M}}$have strong amalgamation. Our results give rise to new instances of structures that admit invariant measures and structures that do not.

Highlights

Randomness is used to construct objects throughout mathematics, and structures resulting from symmetric random constructions often exhibit structural regularities
In terms of a combinatorial criterion, those countable structures in a countable language that can be built via a random construction that is invariant under reorderings of the elements
The Rado graph R, sometimes known as the ‘random graph’, is the unique countable universal ultrahomogeneous graph [53]. It is the Fraısselimit of the class of finite graphs, with a first-order theory characterized by so-called ‘extension axioms’ that have a simple syntactic form. It is the classic example of a countable structure that has a symmetric probabilistic construction, namely, the countably infinite version of the Erdos–Renyi random graph process introduced by Gilbert [24] and Erdos and Renyi [21]

Summary

Introduction

Randomness is used to construct objects throughout mathematics, and structures resulting from symmetric random constructions often exhibit structural regularities. Properties (1) and (2) from Theorem 1.1 are equivalent to the following: (3) There is some N ∈ StrL that has trivial group-theoretic definable closure and is such that there is an Aut(N )-invariant probability measure on StrL concentrated on the set of elements of StrL that are isomorphic to M.

Objectives

Results

Conclusion