QUASI-INVARIANT MEASURES CONCENTRATING ON COUNTABLE STRUCTURES

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.