Discoveries / logic-foundations

Equivalence of generic stability notions for Keisler measures

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.