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Stationarity for fim measures

Prove that if μ ∈ 𝔐_x(U) is a frequency interpretation (fim) measure over a small model M and A is a small set with M ⊆ A ⊆ U, then any A‑invariant, Borel‑definable measure ν ∈ 𝔐_x(U) satisfying ν|_A = μ|_A must equal μ.

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Background

Fim measures generalize generically stable measures outside NIP and satisfy strong averaging properties. The paper proves the result under additional assumptions (e.g., definability of ν or fim of all μ{(n)}), but the full stationarity statement remains conjectural.

Establishing this would align the behavior of fim measures with the stationarity of generically stable types and solidify the measure‑theoretic analogue of key structural properties.

References

\begin{conjecture}\label{conj: stat for fim measures} Let $\mu \in \mathfrak{M}x(U)$ be fim\ over $M \prec U$ and let $A$ be a small set with $M \subseteq A \subseteq U$. If $\nu \in \mathfrak{M}{x}(\mathcal{U})$ is $A$-invariant, Borel-definable, and $\nu|{A} = \mu|{A}$, then $\mu = \nu$.\end{conjecture}

— Definable convolution and idempotent Keisler measures III. Generic stability, generic transitivity, and revised Newelski's conjecture (2406.00912 - Chernikov et al., 3 Jun 2024) in Section 3.2 (Fim measures)