Automorphism invariant measures and weakly generic automorphisms

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