For any group satisfying a suitable chain condition, we construct a finitely additive measure on it that is invariant under certain actions.

In this paper we consider classes of finite structures where we have good control over the sizes of the definable sets. The motivating example is the class of finite fields: it was shown in [1] that for any formulain the language of rings, there are finitely many pairs (d,μ) ∈ω×Q>0so that in any finite fieldFand for any ā ∈Fmthe size |ø(Fn,ā)| is “approximately”μ|F|d. Essentially this is a generalisation of the classical Lang-Weil estimates from the category of varieties to that of (...) the first-order-definable sets. (shrink)

For any group satisfying a suitable chain condition, we construct a finitely additive measure on it that is invariant under certain actions.