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We describe a construction process of a relevant measure in any non-emptycompact metric space. This probability measure has invariance properties withrespect to isometric maps defined on open sets. These properties imply thatthis measure is an appropriate generalisation of the Lebesgue one. Resultsabout its uniqueness are showed, and applications and complementary propertiesare quickly studied. Peculiarly, we show an equivalence result in a generalframework linked with the Krylov-Bogolioubov theorem.

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