Metrically invariant measures on locally homogeneous spaces and hyperspaces

Abstract

We compare different invariance concepts for a Borel measure μ on a metric space, μ is called open-invariant if open isometric sets have equal measure, metrically invariant if isometric Borel sets have equal measure, and strongly invariant if any non-expansive image of A has measuure < μ(A).On common hyperspaces of compact and compact convex sets there are no metrically invariant measures.A locally compact metric space is called locally homogeneous if any two points have isometric neighbourhoods, the isometry transforming one point into the other.On such a space there is a unique open-invariant measure, and this measure is even strongly invariant.