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

TL;DR

This paper introduces a new measure construction in compact metric spaces with invariance properties, generalizing Lebesgue measure, and establishes an equivalence related to the Krylov-Bogolioubov theorem.

Contribution

It presents a novel measure construction with invariance properties and proves an equivalence result connecting it to the Krylov-Bogolioubov theorem.

Findings

The measure generalizes Lebesgue measure in compact metric spaces.

Uniqueness of the measure is established.

An equivalence result related to invariant probabilities is demonstrated.

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.