New existence theorems in measure theory and equivalence results for the existence of invariant probabilities

Abstract

We describe a construction process of a relevant measure in any non-empty compact metric space. This probability measure has invariance properties with respect to isometric maps defined on open sets. These properties imply that this measure is an appropriate generalisation of the Lebesgue one. Results about its uniqueness are showed, and applications and complementary properties are quickly studied. Peculiarly, we show an equivalence result in a general framework linked with the Krylov-Bogolioubov theorem.