Let $\mathcal{A}$ be a countable ℵ₀-homogeneous structure. The primary motivation of this work is to study different amenability properties of (subgroups of) the automorphism group $Aut(\mathcal{A})$ of $\mathcal{A}$; the secondary motivation is to study the existence of weakly generic automorphisms of $\mathcal{A}$. Among others, we present sufficient conditions implying the existence of automorphism invariant probability measures on certain subsets of A and of $Aut(\mathcal{A})$; we also present sufficient conditions implying that the theory of $\mathcal{A}$ is amenable. More concretely, we show that if the set of locally finite automorphisms of $\mathcal{A}$ is dense (in particular, if $\mathcal{A}$ has weakly generic tuples of automorphisms of arbitrary finite length), then there exists a finitely additive probability measure μ on the subsets of $\mathcal{A}$ definable with parameters such that μ is invariant under $Aut(\mathcal{A})$. Moreover, if $\mathcal{A}$ is saturated and the set of its locally finite automorphisms is dense (in particular, if $\mathcal{A}$ is saturated and has weak generics), then the theory of $\mathcal{A}$ is amenable.

中文翻译：

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自同构不变测度和弱泛型自同构

让$\mathcal{一个}$是一个可数的 ℵ ₀ -齐次结构。这项工作的主要动机是研究自同构群（子群）的不同顺应性性质$自动(\mathcal{一个})$的$\mathcal{一个}$; 第二个动机是研究弱泛型自同构的存在$\mathcal{一个}$. 除其他外，我们提出了充分条件，暗示在A的某些子集上存在自同构不变概率测度和$自动(\mathcal{一个})$; 我们还提出了充分条件，暗示$\mathcal{一个}$是顺从的。更具体地说，我们证明了如果$\mathcal{一个}$是稠密的（特别是，如果$\mathcal{一个}$具有任意有限长度的自同构的弱泛型元组），那么在$\mathcal{一个}$可以用参数定义，使得 μ 在下是不变的$自动(\mathcal{一个})$. 此外，如果$\mathcal{一个}$是饱和的，并且它的局部有限自同构集是稠密的（特别是，如果$\mathcal{一个}$是饱和的并且具有弱泛型），那么理论$\mathcal{一个}$是顺从的。