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: $(i)$ $\mu$ is a frequency interpretation measure (fim) over $M$; $(ii)$ $\mu$ is definable over $M$ and its canonical random extension $r_{\mu}$ is generically stable over $M^{\Omega}$; $(iii)$ $\mu$ 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.

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: $(i)$ $\mu$ is a frequency interpretation measure (fim) over $M$; $(ii)$ $\mu$ is definable over $M$ and its canonical random extension $r_{\mu}$ is generically stable over $M^{\Omega}$; $(iii)$ $\mu$ 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.