Metrically invariant measures on locally homogeneous spaces and hyperspaces (Q1083570)

We compare different invariance concepts for a Borel measure \(\mu\) on a metric space. \(\mu\) 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 measure \(\leq \mu (A)\). 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.