Uniformly countably additive families of measures and group invariant measures

The extension of finitely additive measures that are invariant under a group permutations or mappings has already been widely studied. We have dealt with this problem in [15] and [16] from the point of view of Hahn-Banach’s theorem and von Neumann’s measurable groups theory. In this paper we construct countably additive measures from a close point of view, different to that of Haar’s Measure Theory.