Automorphism invariant measures and weakly generic automorphisms

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