Sandwich-type theorems and applications to invariant measures

In this paper we deal with sandwich and Hahn-Banach-type theorems and results about existence of measures, invariant with respect to an Abelian group of transformations. A classical example shows that such theorems may be false, when the involved transformations do not commute. Among the various applications, we study a concept of coherence, related to de Finetti's, and deduce necessary and sufficient conditions for the existence of finitely additive invariant measures, possibly unbounded. Other consequences are concerned with translation-invariant asymptotic densities.