Invariant Measures on Products and on the Space of Linear Orders

Jahel–Tsankov prove the exact dichotomy the model states: for a transitive, ω-categorical structure with no algebraicity and weak elimination of imaginaries, Aut(M) acting on LO(M) either has a fixed point (a definable order) or else the uniform/Glasner–Weiss measure μu is the unique invariant probability. Their Theorem 4.1 states precisely this and the proof proceeds via (i) a de Finetti–type independence theorem (Theorem 3.4, Corollary 3.5), (ii) defining η^τ_a = P(c<a | F_a) which are i.i.d. for each 2-type τ (Lemma 4.2), (iii) an alternating τ-path connectivity lemma (Lemma 4.5) leading to recovery of the order from i.i.d. non-atomic labels (Lemma 4.6, Lemma 4.7), and (iv) an atomic-case conditioning that yields an ergodic Dirac measure at a definable order (Lemmas 4.9–4.11). These are exactly the steps articulated in the model’s solution; the only minor nit is that the model’s outline momentarily conflates the original ergodic μ with the Dirac measure obtained after iterative conditioning, whereas the paper constructs a new ergodic measure ν that is Dirac. Overall, the proofs align essentially verbatim. See Theorem 4.1 and Lemmas 4.2, 4.5–4.7, 4.9–4.11 in the PDF .