Work

Quasi-invariant Measures Concentrating on Countable Structures

Year: 2025

Type: article

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 $\mathcal {L}$ -structures $\mathcal {N}$ 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 byproduct 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.

Source: Journal of Symbolic Logic

DOI: 10.1017/jsl.2025.10133

Authors: CLINTON CONLEY, COLIN JAHEL, ARISTOTELIS PANAGIOTOPOULOS

Institutions: Carnegie Mellon University, American University of Beirut, University of Vienna

Language: English

Topic: Advanced Topology and Set Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,550

Funders: National Science Foundation, Deutsche Forschungsgemeinschaft