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.
