Quasi-invariant measures concentrating on countable structures

TL;DR

This paper characterizes countable structures supporting quasi-invariant probability measures, extending previous work on invariant measures by identifying those not 'highly algebraic' as the key class.

Contribution

It introduces a new characterization of structures supporting quasi-invariant measures, generalizing the known invariant measure case and linking measure properties to algebraic complexity.

Findings

Structures supporting quasi-invariant measures are characterized as not 'highly algebraic'

Any isomorphism class with a quasi-invariant measure admits one with continuous Radon--Nikodym cocycles

Extension of invariant measure characterization to quasi-invariant measures

Abstract

Countable $\mathcal{L}$-structures $\mathcal{N}$ whose isomorphism class supports a permutation invariant probability measure in the logic action have been characterized by Ackerman-Freer-Patel to be precisely those $\mathcal{N}$ which have no algebraicity. Here we characterize those countable $\mathcal{L}$-structure $\mathcal{N}$ whose isomorphism class supports a quasi-invariant probability measure. These turn out to be precisely those $\mathcal{N}$ which are not "highly algebraic" -- we say that $\mathcal{N}$ is highly algebraic if outside of every finite $F$ there is some $b$ and a tuple $\bar{a}$ disjoint from $b$ so that $b$ has a finite orbit under the pointwise stabilizer of $\bar{a}$ in $\mathrm{Aut}(\mathcal{N})$. As a bi-product of our proof we show that whenever the isomorphism class of $\mathcal{N}$ admits a quasi-invariant measure, then it admits one with continuous Radon--Nikodym cocycles.