论一阶顺从性
On first order amenability
AI总结：
本文引入一阶理论的一阶顺从性概念，研究其基本性质与等价条件，证明顺从理论是G-紧的，推广了ω-范畴理论的相关结论，还得到极端顺从性的平行结果。
摘要：
我们引入一阶顺从性的概念，将其作为一阶理论T的一个性质：任意可能含无穷多个变量的空集上的完全型，都可扩张为同变量下的自同构不变全局Keisler测度。当T可数时，T的顺从性可由T的所有足够大的ℵ₀齐次可数模型M的自同构群Aut(M)（拓扑群）的顺从性推出，但前者的约束条件远为宽松。首先，我们研究了顺从理论的基本性质，给出了多个等价条件。随后，我们应用来自E. Hrushovski、K. Krupiński、A. Pillay所著《Amenability, connected components, and definable actions》中的稳定子定理的一个版本，证明了若T是顺从的，则T是G-紧的，即空集上的Lascar强型与Kim-Pillay强型重合。该结论推广并本质拓展了K. Krupiński、A. Pillay在《Amenability, definable groups, and automorphism groups》中通过不同方法针对ω-范畴理论证明的类似结果。在顺从性由空集可定义全局Keisler测度见证的特殊情形（例如顺从的ω-范畴理论即属此类）中，我们还基于连续逻辑中的稳定性给出了另一种证明。针对极端顺从性的概念，我们也得到了平行但更易证明的结果。
英文摘要：
We introduce the notion of first order amenability, as a property of a first order theory $T$: every complete type over $\emptyset$, in possibly infinitely many variables, extends to an automorphism-invariant global Keisler measure in the same variables. Amenability of $T$ follows from amenability of the (topological) group $Aut(M)$ for all sufficiently large $\aleph_{0}$-homogeneous countable models $M$ of $T$ (assuming $T$ to be countable), but is radically less restrictive. First, we study basic properties of amenable theories, giving many equivalent conditions. Then, applying a version of the stabilizer theorem from [Amenability, connected components, and definable actions; E. Hrushovski, K. Krupiński, A. Pillay], we prove that if $T$ is amenable, then $T$ is G-compact, namely Lascar strong types and Kim-Pillay strong types over $\emptyset$ coincide. This extends and essentially generalizes a similar result proved via different methods for $ω$-categorical theories in [Amenability, definable groups, and automorphism groups; K. Krupiński, A. Pillay] . In the special case when amenability is witnessed by $\emptyset$-definable global Keisler measures (which is for example the case for amenable $ω$-categorical theories), we also give a different proof, based on stability in continuous logic. Parallel (but easier) results hold for the notion of extreme amenability.
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