A compact space with a measure that knows which sets are homeomorphic

Abstract

We construct a compact homogeneous space bH which has a Borel measure μ which knows which sets are homeomorphic: if X and Y are homeomorphic Borel sets then μ(X) = μ(Y) , and, as a partical converse, if X and Y are open and μ(X) = μ(Y) and X and Y are both compact or both noncompact, then X and Y are homeomorphic. In particular, μ is nonzero and invariant under all autohomeomorphisms; it turns out that up to a multiplicative constant μ is unique with respect to these properties. bH is constructed as an easy to visualize compactification of a very special sub group H of the circle group T; the Haar measure μ on T induces μ and also induces a measure μ on H which knows which subsets of H are homeomorphic.