Submission date: 14 August 2024

Abstract:

Countable ℒ-structures 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 𝒩 which have no algebraicity. Here we characterize those countable ℒ-structure 𝒩 whose isomorphism class supports a quasi-invariant probability measure. These turn out to be precisely those 𝒩 which are not “highly algebraic” -- we say that 𝒩 is highly algebraic if outside of every finite F there is some b and a tuple ā disjoint from b so that b has a finite orbit under the pointwise stabilizer of ā in Aut(𝒩). As a bi-product of our proof we show that whenever the isomorphism class of 𝒩 admits a quasi-invariant measure, then it admits one with continuous Radon--Nikodym cocycles.

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Full text arXiv 2408.07454: pdf, ps.