Quasi-invariant measures concentrating on countable structures

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.

💡 Research Summary

The paper investigates when the isomorphism class of a countable L‑structure N supports a probability measure that is quasi‑invariant under the natural logic action of the full symmetric group Sym(ℕ). Building on the earlier work of Ackerman, Freer, and Patel, which characterized invariant random structures via the “no‑algebraicity” condition, the authors introduce a new algebraic notion called “highly algebraic”. A structure N is highly algebraic if for every finite set F⊂dom(N) there exist an element b and a tuple ā, disjoint from b, such that b has a finite orbit under the pointwise stabilizer Aut(N)_{ā}. This condition captures a strong form of local algebraicity that persists everywhere outside any finite set.

The main result (Theorem 2) is a precise dichotomy: a countable structure N is quasi‑random (i.e., its isomorphism class carries a Sym(ℕ)‑quasi‑invariant probability measure) if and only if N is not highly algebraic. The forward direction (highly algebraic ⇒ no quasi‑invariant measure) is proved via Lemma 5, which constructs, for any compact subset K of the orbit X=