Equivalence of generic stability notions for Keisler measures

Given a first-order theory $T$ (in discrete or continuous logic) and a Borel-definable global Keisler measure $\\mu$ in $T$, we show that the following conditions are equivalent: $(i)$ $\\mu$ is a frequency interpretation measure; $(ii)$ $\\mu$ is definable and its canonical “random extension” $r_{\\mu}$ is generically stable in the randomization theory $T^{R}$; $(iii)$ $\\mu$ is “self-averaging”. This result establishes a robust notion of generic stability for Keisler measures, which resolves a long-term research objective from previous work. The implications $(i)\\Rightarrow(ii)\\Rightarrow(iii)$ were previously established by the authors (for $T$ discrete). The primary focus of this paper is the reverse implications $(iii)\\Rightarrow(ii)\\Rightarrow(i)$, which we obtain through the use of AI models.

Let $T$ be a complete first-order theory in discrete or continuous logic, let $M\\prec\\mathcal{U}$, and let $\\mu\\in\\mathfrak{M}_{x}(\\mathcal{U})$ be Borel-definable over $M$. The paper proves that the following three conditions are equivalent:\n\n$(i)$ $\\mu$ is a frequency interpretation measure (fim) over $M$;\n\n$(ii)$ $\\mu$ is definable over $M$ and its canonical random extension $r_{\\mu}$ is generically stable over $M^{\\Omega}$;\n\n$(iii)$ $\\mu$ is self-averaging over $M$. The new work proves the reverse implications $(iii)\\Rightarrow(ii)\\Rightarrow(i)$ and extends the characterization to continuous logic. The authors therefore make the equivalent conditions into a definitive definition of generic stability for Keisler measures. The paper also proves a further characterization in terms of an order-property condition and derives consequences for closure under Morley products.